Question 1

How many terms are there in the G.P. 3, 6, 12, 24, ... , 384?

Accepted Answer

Here a = 3 and r = 6 = 2. Let the number of terms be n. 3 Then, tn = 384 ⟹ arn-1 = 384 ⟹ 3 x 2n - 1 = 384 ⟹ 2n-1 = 128 = 27 ⟹ n - 1 = 7 ⟹ n = 8 ∴ Number of terms = 8.

Question 2

(963 + 476)2 + (963 - 476)2 =? (963 x 963 + 476 x 476)

Accepted Answer

Given Exp. = (a + b)2 + (a - b)2 = 2(a2 + b2) = 2 (a2 + b2) (a2 + b2)

Question 3

Which of the following number is divisible by 24?

Accepted Answer

24 = 3 x8, where 3 and 8 co-prime. Clearly, 35718 is not divisible by 8, as 718 is not divisible by 8. Similarly, 63810 is not divisible by 8 and 537804 is not divisible by 8. Consider option (D), Sum of digits = (3 + 1 + 2 + 5 + 7 + 3 + 6) = 27, which is divisible by 3. Also, 736 is divisible by 8. ∴ 3125736 is divisible by (3 x 8), i.e., 24.

Question 4

If the number 5 * 2 is divisible by 6, then * =?

Accepted Answer

6 = 3 x 2. Clearly, 5 * 2 is divisible by 2. Replace * by x Then, (5 + x + 2) must be divisible by 3. So, x = 2.

Question 5

On dividing a number by 5, we get 3 as remainder. What will the remainder when the square of the this number is divided by 5?

Accepted Answer

Let the number be x and on dividing x by 5, we get k as quotient and 3 as remainder. ∴ x = 5k + 3 ⟹ x2 = (5k + 3)2 = (25k2 + 30k + 9) = 5(5k2 + 6k + 1) + 4 ∴On dividing x2 by 5, we get 4 as remainder.

Question 6

35 + 15 x 1.5 =?

Accepted Answer

Given Exp. = 35 + 15 x 3 = 35 + 45 = 35 + 22.5 = 57.5 2 2

Question 7

(12 + 22 + 32 + ... + 102) =?

Accepted Answer

We know that (12 + 22 + 32 + ... + n2) = 1 n(n + 1)(2n + 1) 6 Putting n = 10, required sum = ❨ 1 x 10 x 11 x 21 ❩ = 385 6

Question 8

8597 -? = 7429 - 4358

Accepted Answer

7429          Let 8597 - x = 3071
-4358          Then,      x = 8597 - 3071
 ----                       = 5526
 3071
 ----

Question 9

A number when divide by 6 leaves a remainder 3. When the square of the number is divided by 6, the remainder is:

Accepted Answer

Let x = 6 q + 3. Then, x2 = (6q + 3)2 = 36q2 + 36q + 9 = 6(6q2 + 6q + 1) + 3 Thus, when x2 is divided by 6, then remainder = 3.

Question 10

287 x 287 + 269 x 269 - 2 x 287 x 269 =?

Accepted Answer

Given Exp. = a2 + b2 - 2ab, where a = 287 and b = 269 = (a - b)2 = (287 - 269)2 = (182) = 324

Question 11

On dividing 2272 as well as 875 by 3-digit number N, we get the same remainder. The sum of the digits of N is:

Accepted Answer

Clearly, (2272 - 875) = 1397, is exactly divisible by N. Now, 1397 = 11 x 127 ∴ The required 3-digit number is 127, the sum of whose digits is 10.

Question 12

If x and y are positive integers such that (3x + 7y) is a multiple of 11, then which of the following will be divisible by 11?

Accepted Answer

By hit and trial, we put x = 5 and y = 1 so that (3 x + 7 y ) = (3 x 5 + 7 x 1) = 22, which is divisible by 11. ∴ (4x + 6y) = ( 4 x 5 + 6 x 1) = 26, which is not divisible by 11; (x + y + 4 ) = (5 + 1 + 4) = 10, which is not divisible by 11; (9x + 4y) = (9 x 5 + 4 x 1) = 49, which is not divisible by 11; (4x - 9y) = (4 x 5 - 9 x 1) = 11, which is divisible by 11.

Question 13

The smallest prime number is:

Accepted Answer

The smallest prime number is 2.

Question 14

9548 + 7314 = 8362 + (?)

Accepted Answer

9548          16862 = 8362 + x 
+ 7314          x  = 16862 - 8362 
 -----             = 8500 
 16862
 -----

Question 15

Which of the following numbers will completely divide (461 + 462 + 463 + 464)?

Accepted Answer

(4 61 + 4 62 + 4 63 + 4 64 ) = 4 61 x (1 + 4 + 4 2 + 4 3 ) = 4 61 x 85 = 460 x (4 x 85) = (460 x 340), which is divisible by 10.

Question 16

The sum of how many terms of the series 6 + 12 + 18 + 24 + ... is 1800?

Accepted Answer

This is an A.P. in which a = 6, d = 6 and S n = 1800 Then, n [2a + (n - 1)d] = 1800 2 ⟹ n [2 x 6 + (n - 1) x 6] = 1800 2 ⟹ 3n (n + 1) = 1800 ⟹ n(n + 1) = 600 ⟹ n2 + n - 600 = 0 ⟹ n2 + 25n - 24n - 600 = 0 ⟹ n(n + 25) - 24(n + 25) = 0 ⟹ (n + 25)(n - 24) = 0 ⟹ n = 24 Number of terms = 24.

Question 17

666 / 6 / 3 =?

Accepted Answer

Given Exp. = 666 x 1 x 1 = 37 6 3

Question 18

Which one of the following numbers is completely divisible by 99?

Accepted Answer

99 = 11 x 9, where 11 and 9 are co-prime. By hit and trial, we find that 114345 is divisibleby 11 as well as 9. So, it is divisible by 99.

Question 19

(51+ 52 + 53 + ... + 100) =?

Accepted Answer

This is an A.P. in which a = 51, l = 100 and n = 50. ∴Sum = n (a + l) = 50 x (51 + 100) = (25 x 151) = 3775. 2 2

Question 20

The difference between the place value and the face value of 7 in the numeral 967452 is

Accepted Answer

(Place value of 7)-(face value of 7) =7000-7=6993.

Numbers Questions