APTITUDE QUESTIONS

A canteen requires 42 dozen eggs per week. How many eggs will it require for 70 days? 6048 5040 3024 3628

ChatGPT said:

To find how many eggs the canteen requires for 70 days, let's break it down step by step:

Step 1: Convert dozens to eggs

The canteen requires 42 dozen eggs per week.

1 dozen = 12 eggs
So, 42 dozen = $42 \times 12 = 504$ eggs per week

Step 2: Convert weeks to days

1 week = 7 days
So, daily requirement = $\frac{504}{7} = 72$ eggs per day

Step 3: Calculate for 70 days

$72 \times 70 = 5040$ eggs

✅ Final Answer: 5040

Correct option: 5040 ✅

You said:

A number becomes perfect square if it is multiplied by three. Even if the number is divided by 3 it remains a perfect square. Find out that number. 36 16 9 27

ChatGPT said:

We are given:

A number becomes a perfect square if it is multiplied by 3.
Even if it is divided by 3, it remains a perfect square.

Let the number be $x$ .

So:

$x \times 3$ is a perfect square
$\frac{x}{3}$ is also a perfect square

That means:

x \times 3 = a^2 \quad \text{(1)}\\ \frac{x}{3} = b^2 \quad \text{(2)}

From equation (1):

x = \frac{a^2}{3}

From equation (2):

x = 3b^2

Equating the two expressions for $x$ :

\frac{a^2}{3} = 3b^2 \\ a^2 = 9b^2 \\ a = 3b

Now plug back into one of the expressions:

x = 3b^2 \Rightarrow \text{Try } b = 1 \Rightarrow x = 3(1)^2 = 3 \quad \text{(No, not a perfect square when multiplied)} \\ b = 2 \Rightarrow x = 3(4) = 12 \quad \text{→ } 12 × 3 = 36 ✔ (perfect square), 12 ÷ 3 = 4 ✔ (perfect square) ✅ So, the number is **12** But **12 is not in the options**. Let's look at the options: - **36** → 36 × 3 = 108 ❌ not a perfect square 36 ÷ 3 = 12 ❌ not a perfect square - **16** → 16 × 3 = 48 ❌ 16 ÷ 3 ≈ 5.33 ❌ - **9** → 9 × 3 = 27 ❌ 9 ÷ 3 = 3 ❌ - **27** → 27 × 3 = 81 ✔ (perfect square) 27 ÷ 3 = 9 ✔ (perfect square) ✅ So the correct number is **27** ### ✅ Final Answer: **27**

You said:

A number of children are standing in a circle. They are evenly spaced and the 6th child is directly opposite the 20th child. How many children are there altogether ? 28 30 26 16

We are told:

A group of children are evenly spaced in a circle.
The 6th child is directly opposite the 20th child.

Step 1: Understand what "directly opposite" means

In a circle, if one child is directly opposite another, then the number of children between them is half the total number of children.

So the number of children between child 6 and child 20 is half of the total number.

Let’s compute the distance between child 6 and 20:

|20 - 6| = 14

So, 14 children between 6 and 20, when moving around the circle.

This 14 is half of the total number of children in the circle:

\text{Total} = 14 \times 2 = \boxed{28}

✅ Final Answer: 28

You said:

If 17^200 is divided by 18, the remainder is 17 16 2 1

We are asked to find the remainder when $17^{200}$ is divided by 18.

This is a modular arithmetic problem:

17^{200} \mod 18

Step 1: Notice that 17 ≡ -1 mod 18

Since:

17 \mod 18 = -1 \quad \text{(because 17 is 1 less than 18)}

So,

17^{200} \equiv (-1)^{200} \mod 18

Since 200 is even:

(-1)^{200} = 1

Therefore,

17^{200} \mod 18 = 1

✅ Final Answer: 1

If 3 2/3 is subtracted from 9 and the difference is multiplied by 450. What is the final answer? 2045 2400 2250 2540

We are given:

If $3 \frac{2}{3}$ is subtracted from 9, and the difference is multiplied by 450, what is the final answer?

Step 1: Convert mixed fraction to improper fraction

3 \frac{2}{3} = \frac{11}{3}

Step 2: Subtract from 9

9 - \frac{11}{3} = \frac{27}{3} - \frac{11}{3} = \frac{16}{3}

Step 3: Multiply the result by 450

\frac{16}{3} \times 450 = \frac{16 \times 450}{3} = \frac{7200}{3} = 2400

✅ Final Answer: 2400

The cost of 8 chairs and 20 tables is Rs 3960. What will be the cost of 28 chairs and 70 tables? Rs 13860 Rs 12620 Rs 13240 Rs 12820

We are told:

The cost of 8 chairs and 20 tables = Rs 3960
We are to find the cost of 28 chairs and 70 tables

Step 1: Observe the ratio

Let’s check if we can scale up the quantities directly:

The new quantities:

Chairs: $28 = 8 \times 3.5$
Tables: $70 = 20 \times 3.5$

So, both quantities are multiplied by 3.5

Step 2: Multiply total cost by 3.5

3960 \times 3.5 = ?

Break it down:

3960 \times 3.5 = (3960 \times 3) + (3960 \times 0.5) \\ = 11,880 + 1980 = 13,860

✅ Final Answer: Rs 13,860

The sum of all even natural numbers less than 100 is : 2272 2352 2450 2468

We are asked to find the sum of all even natural numbers less than 100.

Step 1: Identify the even numbers less than 100

Even numbers less than 100:
$2, 4, 6, ..., 98$

This is an arithmetic progression (AP):

First term $a = 2$
Last term $l = 98$
Common difference $d = 2$

Step 2: Use the AP sum formula

\text{Number of terms}, n = \frac{(l - a)}{d} + 1 = \frac{98 - 2}{2} + 1 = 49 + 1 = 50

\text{Sum} = \frac{n}{2}(a + l) = \frac{50}{2}(2 + 98) = 25 \times 100 = 2500

✅ Final Answer: 2500

None of the options you listed (2272, 2352, 2450, 2468) are correct. The correct sum is 2500.

The sum of three consecutive numbers is 174. What is the product of the smallest and the largest number? 3306 3432 3422 3363

We are told:

The sum of three consecutive numbers is 174.
Find the product of the smallest and largest number.

Step 1: Let the three consecutive numbers be:

x - 1,\ x,\ x + 1

Their sum is:

(x - 1) + x + (x + 1) = 3x

Set this equal to 174:

3x = 174 \Rightarrow x = \frac{174}{3} = 58

So the three numbers are:

57,\ 58,\ 59

Step 2: Product of smallest and largest:

57 \times 59

Use the identity:

(a - 1)(a + 1) = a^2 - 1

Here, $a = 58$ , so:

57 \times 59 = 58^2 - 1 = 3364 - 1 = 3363

✅ Final Answer: 3363

Three-fourths of a number is 60 more than its one-third, the number is 144 108 84 160

We are told:

Three-fourths of a number is 60 more than its one-third.

Let the number be $x$ .

Step 1: Set up the equation

\frac{3}{4}x = \frac{1}{3}x + 60

Step 2: Eliminate fractions (multiply entire equation by 12)

12 \times \left( \frac{3}{4}x \right) = 12 \times \left( \frac{1}{3}x + 60 \right)

9x = 4x + 720

Step 3: Solve for $x$

9x - 4x = 720 \Rightarrow 5x = 720 \Rightarrow x = \frac{720}{5} = 144

✅ Final Answer: 144

What is the least number to be added to 920 to make it a perfect square ? 41 49 39 31

We need to find the least number to be added to 920 to make it a perfect square.

Step 1: Find the square root of 920 (approximate)

\sqrt{920} \approx 30.33

The next perfect square after $30^2 = 900$ is:

31^2 = 961

Step 2: Calculate the difference

961 - 920 = 41

✅ Final Answer: 41

When an amount of Rs 62396 is divided equally amongst 38 people what amount will each person get? Rs.1612 Rs.1584 Rs.1556 Rs.1642

Given:

Total amount = Rs 62,396
Number of people = 38

Step 1: Divide the total amount by the number of people

\frac{62396}{38} = ?

Let's calculate:

$38 \times 1600 = 60,800$
Difference = $62,396 - 60,800 = 1,596$
$38 \times 40 = 1,520$
Remaining = $1,596 - 1,520 = 76$
$38 \times 2 = 76$

So total:

1600 + 40 + 2 = 1642

✅ Final Answer: Rs. 1642

Which is the largest 4 digit number by exactly divisible by 88 ? 9768 9988 9944 8888

We need to find the largest 4-digit number exactly divisible by 88 among the options:

9768
9988
9944
8888

Step 1: Check divisibility by 88

Since $88 = 8 \times 11$ , the number must be divisible by both 8 and 11.

Check each option:

1) 9988

Divisible by 8?
Check last 3 digits: 988
$988 \div 8 = 123.5$ (Not divisible)

So, No

2) 9944

Divisible by 8?
Last 3 digits: 944
$944 \div 8 = 118$ (Divisible)
Divisible by 11?
Check difference of sum of alternate digits:
Digits: 9 9 4 4
Sum of 1st and 3rd digits = 9 + 4 = 13
Sum of 2nd and 4th digits = 9 + 4 = 13
Difference = 13 - 13 = 0 (Divisible by 11)

So, Yes

3) 9768

Divisible by 8?
Last 3 digits: 768
$768 \div 8 = 96$ (Divisible)
Divisible by 11?
Digits: 9 7 6 8
Sum of 1st and 3rd digits = 9 + 6 = 15
Sum of 2nd and 4th digits = 7 + 8 = 15
Difference = 0 (Divisible by 11)

So, Yes

4) 8888

Divisible by 8?
Last 3 digits: 888
$888 \div 8 = 111$ (Divisible)
Divisible by 11?
Digits: 8 8 8 8
Sum of 1st and 3rd digits = 8 + 8 = 16
Sum of 2nd and 4th digits = 8 + 8 = 16
Difference = 0 (Divisible by 11)

So, Yes

Step 2: Find the largest among 9944, 9768, 8888

9944
9768
8888

Largest is 9944

✅ Final Answer: 9944

Find the greatest number of three digits, which is exactly divisible by 35. 900 980 960 970

We are asked to find the greatest 3-digit number divisible by 35.

Step 1: Largest 3-digit number is 999

We want:

\text{Greatest 3-digit number divisible by 35} \leq 999

Step 2: Divide 999 by 35

999 \div 35 \approx 28.54 \Rightarrow \text{Take the integer part: } 28

Now multiply:

35 \times 28 = 980

✅ Final Answer: 980

A number when divided by a divisor, left remainder 23, when twice of the number was divided by the same divisor, remainder was 11. Find the divisor. 34 24 12

We are told:

A number when divided by a divisor, leaves remainder 23
When twice the number is divided by the same divisor, the remainder is 11
We are to find the divisor

Step 1: Let the number be $x$ , and the divisor be $d$

From the first condition:

x \equiv 23 \pmod{d}

From the second condition:

2x \equiv 11 \pmod{d}

Step 2: Use substitution

We already know:

x \equiv 23 \pmod{d} \Rightarrow 2x \equiv 2 \times 23 = 46 \pmod{d}

But we are told:

2x \equiv 11 \pmod{d}

So:

46 \equiv 11 \pmod{d} \Rightarrow 46 - 11 = 35 \equiv 0 \pmod{d}

This means:

d \mid 35

Step 3: Find the divisors of 35

\text{Divisors of 35: } 1,\ 5,\ 7,\ 35

Now check which of these are in the options:
34, 24, 12 → None of them divide 35.

Looks like there's a mistake in the options provided.

✅ Correct divisor is 35

✅ Final Answer: 35 (not among the options given)

A painter has painted one-third of a rectangular wall which is ten feet high. When she has painted another 75 square feet of wall, she will be three quarters finished with the job. What is the length (The horizontal dimension) of the wall ? 18 ft 12 ft 9 ft

We are given:

The wall is rectangular, 10 feet high
The painter has painted one-third of the wall already
After painting 75 more square feet, she will have completed three-fourths of the wall

We need to find the length (horizontal dimension) of the wall.

Step 1: Let the length of the wall be $L$ feet

So, total area of the wall:

\text{Area} = 10 \times L

Step 2: Use the given conditions

Painted so far = $\frac{1}{3} \times 10L = \frac{10L}{3}$
After painting 75 more sq ft, total painted = $\frac{10L}{3} + 75$
At that point, the job is three-fourths done:

\frac{10L}{3} + 75 = \frac{3}{4} \times 10L = \frac{30L}{4} = \frac{15L}{2}

Step 3: Set up the equation

\frac{10L}{3} + 75 = \frac{15L}{2}

Multiply through by 6 to eliminate denominators:

6 \left( \frac{10L}{3} + 75 \right) = 6 \cdot \frac{15L}{2} \Rightarrow 20L + 450 = 45L

Step 4: Solve for $L$

450 = 45L - 20L = 25L \Rightarrow L = \frac{450}{25} = 18

✅ Final Answer: 18 ft

A sales man recives c percentage commission on a sale of r rupees. How many rupees does he receive as commission? r/100 cr/100 cr c/100

We are told:

A salesman receives c percent commission on a sale of r rupees.
We are to find how many rupees he receives as commission.

Step-by-step:

Commission is c% of r, which means:

\text{Commission} = \frac{c}{100} \times r = \frac{cr}{100}

✅ Final Answer: cr/100

If the difference of two numbers is 10 and their product is 12, what is the sum of their squares ? 169 181 124 125

We are told:

The difference of two numbers is 10
Their product is 12
We are to find the sum of their squares

Step 1: Let the two numbers be $x$ and $y$

Given:

$x - y = 10$
$xy = 12$
Find: $x^2 + y^2$

Step 2: Use identity:

x^2 + y^2 = (x - y)^2 + 2xy

Substitute values:

$(x - y)^2 = 10^2 = 100$
$2xy = 2 \times 12 = 24$

x^2 + y^2 = 100 + 24 = 124

✅ Final Answer: 124

In a polling booth total number of voters is 1575, of which 0.4 part are male voters. If a candidate gets 0.6 part of male voters and 0.4 part of female voters, then find how many votes did the candidate get ? 576 765 756 657

We are given:

Total number of voters = 1575
0.4 part are male voters
So, 0.6 part are female voters
Candidate gets:
- 0.6 of male votes
- 0.4 of female votes

We are to find how many votes the candidate got.

Step 1: Find number of male and female voters

Male voters = $0.4 \times 1575 = 630$
Female voters = $0.6 \times 1575 = 945$

Step 2: Find how many votes the candidate got

From male voters: $0.6 \times 630 = 378$
From female voters: $0.4 \times 945 = 378$

Step 3: Total votes

378 + 378 = 756

✅ Final Answer: 756

The four integers next lower than 81 and the four next integers higher than 81 are write down and added together, this sum divisible by ? 7 11 13 9

We are given:

Take the four integers lower than 81:
$80, 79, 78, 77$
Take the four integers higher than 81:
$82, 83, 84, 85$

Step 1: Add all 8 numbers:

77 + 78 + 79 + 80 + 82 + 83 + 84 + 85

Break it into two parts:

Lower 4: $77 + 78 + 79 + 80 = 314$
Higher 4: $82 + 83 + 84 + 85 = 334$

\text{Total sum} = 314 + 334 = 648

Step 2: Check which option divides 648

$648 \div 7 = 92.57$ ❌
$648 \div 11 = 58.91$ ❌
$648 \div 13 = 49.84$ ❌
$648 \div 9 = 72$ ✅

✅ Final Answer: 9

The multiplicative inverse of the non-zero rational number x is 1/x x 1 0

The multiplicative inverse of a non-zero rational number $x$ is the number that, when multiplied by $x$ , gives 1.

x \times \frac{1}{x} = 1

✅ Final Answer: 1/x

A number decreases by 20 percent give 48. Find the number. 80 75 50 60

We are told:

A number decreases by 20% and becomes 48
We are to find the original number

Step 1: Let the original number be $x$

20% decrease means:

x - \frac{20}{100}x = 48 \Rightarrow \frac{80}{100}x = 48 \Rightarrow \frac{4}{5}x = 48

Step 2: Solve for $x$

x = \frac{48 \times 5}{4} = \frac{240}{4} = 60

✅ Final Answer: 60

Find the l.c.m of 10, 12, 15 and 20. 120 60 240 150

We are asked to find the LCM of 10, 12, 15, and 20.

Step 1: Prime factorizations

$10 = 2 \times 5$
$12 = 2^2 \times 3$
$15 = 3 \times 5$
$20 = 2^2 \times 5$

Step 2: Take the highest powers of all primes involved:

$2^2$ (from 12 or 20)
$3$ (from 12 or 15)
$5$ (from 10, 15, or 20)

\text{LCM} = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60

✅ Final Answer: 60

If the cost of a book worth Rs. 50 is increased by Rs. 25, the rate of increase is? 20 percent 25 percent 50 percent 10 percent

We are told:

Original cost of the book = Rs. 50
Increase in cost = Rs. 25
Find the rate of increase (percentage)

Step 1: Use the formula for percentage increase

\text{Rate of increase} = \left(\frac{\text{Increase}}{\text{Original cost}}\right) \times 100

= \frac{25}{50} \times 100 = 0.5 \times 100 = 50\%

✅ Final Answer: 50 percent

The product of two numbers is 120 and its l.c.m is given 60, find its h.c.f 3 5 4 2

We are given:

Product of two numbers = 120
L.C.M. of the two numbers = 60
Find the H.C.F.

Step 1: Use the relationship between product, LCM, and HCF

\text{Product} = \text{LCM} \times \text{HCF}

Let the HCF be $h$ .

120 = 60 \times h \implies h = \frac{120}{60} = 2

✅ Final Answer: 2

A boy was asked to multiply a certain number by 25. He multiplied it by 52 and got his answer 324 more than the correct one) The number to be multiplied was 12 15 25 52

We are told:

A boy was supposed to multiply a number by 25
He mistakenly multiplied it by 52
His result was 324 more than the correct answer
We are to find the original number

Step 1: Let the number be $x$

Correct result = $25x$
Mistaken result = $52x$

The difference:

52x - 25x = 324 \Rightarrow 27x = 324

Step 2: Solve for $x$

x = \frac{324}{27} = 12

✅ Final Answer: 12

The number 77 is divided into two parts such that H.C. F. of two parts is 11 and their L.C. M. is 110. The smallest of the two parts is 11 22 44 55

We are given:

A number (77) is divided into two parts
Their HCF is 11
Their LCM is 110
We are to find the smaller part

Step 1: Let the two parts be $a$ and $b$

Given:

$a + b = 77$
$\text{HCF}(a, b) = 11$
$\text{LCM}(a, b) = 110$

Also:

a \times b = \text{HCF} \times \text{LCM} = 11 \times 110 = 1210

Step 2: Use system of equations

We now have:

$a + b = 77$
$ab = 1210$

Use the identity:

(a + b)^2 = a^2 + b^2 + 2ab \Rightarrow a^2 + b^2 = (a + b)^2 - 2ab \Rightarrow 77^2 - 2 \times 1210 = 5929 - 2420 = 3509

But that's extra—we can directly solve for a and b using a quadratic:

Let’s assume:

a + b = 77,\quad ab = 1210

Let the numbers be roots of the quadratic:

x^2 - 77x + 1210 = 0

Solve using quadratic formula:

x = \frac{77 \pm \sqrt{77^2 - 4 \cdot 1210}}{2} = \frac{77 \pm \sqrt{5929 - 4840}}{2} = \frac{77 \pm \sqrt{1089}}{2} = \frac{77 \pm 33}{2}

So:

$x = \frac{77 + 33}{2} = 55$
$x = \frac{77 - 33}{2} = 22$

Step 3: Smaller of the two = 22

✅ Final Answer: 22

The smallest number which is divisible by 12, 15 and 20 is a perfect square, is: 400 900 1600 3600

We are asked to find the smallest number which:

Is divisible by 12, 15, and 20
Is also a perfect square

Step 1: Find the LCM of 12, 15, and 20

Prime factorizations:

$12 = 2^2 \times 3$
$15 = 3 \times 5$
$20 = 2^2 \times 5$

Take maximum powers of all primes:

\text{LCM} = 2^2 \times 3 \times 5 = 60

So, the LCM = 60
But 60 is not a perfect square.

Step 2: Find the smallest multiple of 60 that is a perfect square

Try multiples of 60 that are perfect squares:

$60 \times 1 = 60$ ❌
$60 \times 2 = 120$ ❌
$60 \times 3 = 180$ ❌
$60 \times 4 = 240$ ❌
Keep going...

Check:

$60 \times 15 = 900$

Now check if 900 is a perfect square:

\sqrt{900} = 30 \Rightarrow \text{Yes}

Also, check if 900 is divisible by 12, 15, and 20:

$900 \div 12 = 75$ ✅
$900 \div 15 = 60$ ✅
$900 \div 20 = 45$ ✅

✅ Final Answer: 900

Find the number of factors of 1440. 32 35 36 40

To find the number of factors of 1440, follow these steps:

Step 1: Prime factorize 1440

Let's divide step by step:

1440 \div 2 = 720 720 \div 2 = 360 360 \div 2 = 180 180 \div 2 = 90 90 \div 2 = 45 45 \div 3 = 15 15 \div 3 = 5 5 \div 5 = 1

So:

1440 = 2^5 \times 3^2 \times 5^1

Step 2: Use the formula for number of factors

If $n = p^a \times q^b \times r^c$ ,
then total number of factors = $(a+1)(b+1)(c+1)$

So:

\text{Number of factors of } 1440 = (5+1)(2+1)(1+1) = 6 \times 3 \times 2 = 36

✅ Final Answer: 36

For a group of 100 candidates, the mean was found to be 40. Later it was discovered that a value of 95 was misread as 54. The correct means is 40.5 39.85 39.8 40.41

We are given:

Mean of 100 candidates = 40
So, total sum = $100 \times 40 = 4000$
But there was an error: 95 was misread as 54
So the sum was short by: $95 - 54 = 41$

Step 1: Corrected total sum

4000 + 41 = 4041

Step 2: Corrected mean

\text{Correct mean} = \frac{4041}{100} = 40.41

✅ Final Answer: 40.41

We are given:

$\frac{1}{8}$ of the pencil is black
$\frac{1}{2}$ of the remaining is white
The rest, 3½ cm, is blue
We are to find the total length of the pencil

Step 1: Let the total length be $x$ cm

Black part = $\frac{1}{8}x$
Remaining = $x - \frac{1}{8}x = \frac{7}{8}x$
White part = $\frac{1}{2} \times \frac{7}{8}x = \frac{7}{16}x$
Blue part = remaining after black and white:

x - \left(\frac{1}{8}x + \frac{7}{16}x\right) = 3.5

Step 2: Simplify the equation

\frac{1}{8}x + \frac{7}{16}x = \frac{2}{16}x + \frac{7}{16}x = \frac{9}{16}x

So:

x - \frac{9}{16}x = \frac{7}{16}x = 3.5

Step 3: Solve for $x$

\frac{7}{16}x = 3.5 = \frac{7}{2} \Rightarrow x = \frac{7}{2} \times \frac{16}{7} = 8

✅ Final Answer: 8 cm

If numerator and denominator are increased by 20 percent and 30 percent respectively the fraction become 9/13. What was the original fraction ? 3/5 5/6 4/5 3/4

We are given:

After increasing the numerator by 20% and denominator by 30%, the fraction becomes $\frac{9}{13}$
We are to find the original fraction

Step 1: Let the original fraction be $\frac{x}{y}$

New numerator = $x + 20\% \text{ of } x = 1.2x$
New denominator = $y + 30\% \text{ of } y = 1.3y$

So:

\frac{1.2x}{1.3y} = \frac{9}{13}

Step 2: Cross multiply

1.2x \cdot 13 = 1.3y \cdot 9 \Rightarrow 15.6x = 11.7y

Divide both sides by 1.3:

12x = 9y \Rightarrow \frac{x}{y} = \frac{9}{12} = \frac{3}{4}

✅ Final Answer: 3/4

In a division sum the divisor is 12 times the quotient and 5 times the remainder. If the remainder is 48, then the dividend is ? 4850 4864 4848 4949

We are given:

Divisor = 12 × Quotient
Divisor = 5 × Remainder
Remainder = 48
Find the Dividend

Step 1: Let

Quotient = $q$
Remainder = $r = 48$
Divisor = $d$

From the problem:

$d = 12q$
$d = 5 \times 48 = 240$

Now solve for $q$ :

12q = 240 \Rightarrow q = \frac{240}{12} = 20

Step 2: Use dividend formula

\text{Dividend} = (\text{Divisor} \times \text{Quotient}) + \text{Remainder} = (240 \times 20) + 48 = 4800 + 48 = 4848

✅ Final Answer: 4848

A number when divided by 296, we find the remainder 75. If the same number is divided by 37 then what will be the remainder 11 8 2 1

We are given:

A number when divided by 296, leaves remainder 75
We are to find the remainder when the same number is divided by 37

Step 1: Let the number be:

N = 296k + 75 \quad \text{(for some integer } k)

We want to find:

N \mod 37 = (296k + 75) \mod 37

Break it into two parts:

N \mod 37 = [(296k) \mod 37 + 75 \mod 37] \mod 37

Step 2: Simplify modulo 37

$296 \div 37 = 8$ , and $37 \times 8 = 296 \Rightarrow 296 \mod 37 = 0$
So $296k \mod 37 = 0$

Now:

N \mod 37 = (0 + 75) \mod 37 = 75 \mod 37

Now divide 75 by 37:

75 = 2 \times 37 + 1 \Rightarrow \text{Remainder} = 1

✅ Final Answer: 1

Find the greatest number which divides 62, 132 and 279 leaving the same remainder in each case. 48 21 7 14

We are asked to find the greatest number that divides 62, 132, and 279 leaving the same remainder in each case.

✅ Key Concept:

If a number leaves the same remainder when dividing multiple numbers, then the differences between those numbers are all divisible by that number.

So, let the required number be $x$ . Then:

$x \mid (132 - 62 = 70)$
$x \mid (279 - 132 = 147)$
$x \mid (279 - 62 = 217)$

So $x$ divides 70, 147, and 217

Step 1: Find the GCD of 70, 147, and 217

GCD(70, 147)

Prime factors:
- $70 = 2 \times 5 \times 7$
- $147 = 3 \times 7 \times 7$
Common factor = 7

Now check:

GCD(7, 217)

$217 \div 7 = 31$ , so 7 divides 217.

✅ So the greatest number dividing all three differences = 7

✅ Final Answer: 7

If N is the smallest perfect square which is exactly divisible by 10, 12 and 25, then the quotient obtained when N is divided by 25 is 36 64 25 16

We are given:

$N$ is the smallest perfect square divisible by 10, 12, and 25
Find the quotient when $N$ is divided by 25

Step 1: Find the LCM of 10, 12, and 25

Prime factorizations:

$10 = 2 \times 5$
$12 = 2^2 \times 3$
$25 = 5^2$

Take the highest powers:

\text{LCM} = 2^2 \times 3 \times 5^2 = 4 \times 3 \times 25 = 300

Step 2: Make $N$ a perfect square and divisible by 300

The prime factorization of 300 is:

300 = 2^2 \times 3^1 \times 5^2

To make $N$ a perfect square, all exponents must be even.

$2^2$ is already even
$3^1$ is odd → multiply by another 3 to get $3^2$
$5^2$ is already even

So, multiply 300 by $3$ to get:

N = 300 \times 3 = 900

$900 = 2^2 \times 3^2 \times 5^2$ , a perfect square.

Step 3: Find the quotient when $N = 900$ is divided by 25

\frac{900}{25} = 36

✅ Final Answer: 36

Monica, Veronica and Rachael begin to jog around a circular stadium. They complete their revolutions in 42 seconds, 56 seconds and 63 seconds respectively. After how many seconds will they be together at the starting point? 336 252 504 Can't be determined

We are given:

Monica completes a revolution in 42 seconds
Veronica completes a revolution in 56 seconds
Rachael completes a revolution in 63 seconds

We need to find after how many seconds they will all meet together at the starting point.

Step 1: Find the LCM of the times

Find $\text{LCM}(42, 56, 63)$ .

Prime factorize each:

$42 = 2 \times 3 \times 7$
$56 = 2^3 \times 7$
$63 = 3^2 \times 7$

Take highest powers:

$2^3 = 8$
$3^2 = 9$
$7^1 = 7$

Calculate LCM:

8 \times 9 \times 7 = 504

✅ Final Answer: 504 seconds

The greatest number of three digits which when divided by 10, 12, 15 leaves remainder 4 in each case, is 996 992 964 984

Given:

We want the greatest 3-digit number $N$ that leaves a remainder 4 when divided by 10, 12, and 15.

Step 1: Let the number be:

N = k \times \text{LCM}(10, 12, 15) + 4

where $k$ is an integer.

Step 2: Find the LCM of 10, 12, and 15

Prime factorization:

$10 = 2 \times 5$
$12 = 2^2 \times 3$
$15 = 3 \times 5$

LCM takes highest powers:

\text{LCM} = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60

Step 3: Express $N$ :

N = 60k + 4

We want the greatest 3-digit number (≤ 999):

60k + 4 \leq 999 \Rightarrow 60k \leq 995 \Rightarrow k \leq \frac{995}{60} = 16.58

So the greatest integer $k = 16$ .

Step 4: Calculate $N$ :

N = 60 \times 16 + 4 = 960 + 4 = 964

✅ Final Answer: 964

The sum of two numbers is 528 and their HCF is 33. How many pairs of such numbers can be? 3 4 1 2

Given:

Sum of two numbers = 528
HCF = 33
Find the number of pairs of such numbers

Step 1: Let the two numbers be $33a$ and $33b$ , where $a$ and $b$ are co-prime (since HCF is factored out).

Step 2: Sum of numbers:

33a + 33b = 528 \Rightarrow 33(a + b) = 528 \Rightarrow a + b = \frac{528}{33} = 16

Step 3: Since $a$ and $b$ are co-prime positive integers with $a + b = 16$ , find the number of such pairs.

We need to find pairs $(a,b)$ such that:

$a + b = 16$
$\gcd(a,b) = 1$
$a, b$ are positive integers

Step 4: List pairs $(a,b)$ with sum 16:

(1,15), (2,14), (3,13), (4,12), (5,11), (6,10), (7,9), (8,8), (9,7), (10,6), (11,5), (12,4), (13,3), (14,2), (15,1)

Step 5: Check which pairs are co-prime:

$\gcd(1,15) = 1$ ✓
$\gcd(2,14) = 2$ ✗
$\gcd(3,13) = 1$ ✓
$\gcd(4,12) = 4$ ✗
$\gcd(5,11) = 1$ ✓
$\gcd(6,10) = 2$ ✗
$\gcd(7,9) = 1$ ✓
$\gcd(8,8) = 8$ ✗

The other pairs are repeats of above (since order reversed).

Step 6: Valid pairs are:

(1,15), (3,13), (5,11), (7,9)

Number of such pairs = 4

✅ Final Answer: 4

Three sets of English, Mathematics and Science books containing 336, 240, 96 books respectively have to be stacked in such a way that all the books are stored subject wise and the height of each stack is the same. Total number of stacks will be 22 14 48 21

We are given:

English books = 336
Mathematics books = 240
Science books = 96

Goal:

Stack the books subject-wise
Each stack has the same height (number of books)
Find the total number of stacks

Step 1: To have stacks of the same height in all subjects, the height must be a common divisor of 336, 240, and 96.

Find the greatest common divisor (GCD) of 336, 240, and 96.

Step 2: Find the GCD

Prime factors:
- $336 = 2^4 \times 3 \times 7$
- $240 = 2^4 \times 3 \times 5$
- $96 = 2^5 \times 3$

Common factors are:

$2^{4}$ (minimum power of 2 common in all)
$3^1$

So,

\text{GCD} = 2^4 \times 3 = 16 \times 3 = 48

Step 3: Number of stacks per subject

English stacks = $\frac{336}{48} = 7$
Math stacks = $\frac{240}{48} = 5$
Science stacks = $\frac{96}{48} = 2$

Step 4: Total number of stacks

7 + 5 + 2 = 14

✅ Final Answer: 14

Find the average of first 41 odd numbers. 43 42 40 41

We want to find the average of the first 41 odd numbers.

Step 1: Recall the first $n$ odd numbers are:

1, 3, 5, \dots, (2n - 1)

For $n = 41$ , the last odd number is:

2 \times 41 - 1 = 81

Step 2: Sum of first $n$ odd numbers:

1 + 3 + 5 + \dots + (2n - 1) = n^2

So, sum of first 41 odd numbers is:

41^2 = 1681

Step 3: Average:

\text{Average} = \frac{\text{Sum}}{\text{Number of terms}} = \frac{1681}{41} = 41