Question 1

The difference of the squares of two consecutive even integers is divisible by which of the following integers?

Accepted Answer

Let the two consecutive even integers be 2 n and (2 n + 2). Then, (2n + 2)2 = (2n + 2 + 2n)(2n + 2 - 2n) = 2(4n + 2) = 4(2n + 1), which is divisible by 4.

Question 2

When a number is divided by 13, the remainder is 11. When the same number is divided by 17, then remainder is 9. What is the number?

Accepted Answer

x = 13 p + 11 and x = 17 q + 9 ∴ 13p + 11 = 17q + 9 ⟹ 17q - 13p = 2 ⟹ q = 2 + 13p 17 The least value of p for which q = 2 + 13p is a whole number is p = 26 17 ∴ x = (13 x 26 + 11) = (338 + 11) = 349

Question 3

The largest 5 digit number exactly divisible by 91 is:

Accepted Answer

Largest 5-digit number = 99999

 91) 99999 (1098
     91
     ---
     899
     819
     ----
      809
      728
      ---
       81
      ---
 
 Required number = (99999 - 81)
                 = 99918.

Question 4

The sum of even numbers between 1 and 31 is:

Accepted Answer

Let S n = (2 + 4 + 6 + ... + 30). This is an A.P. in which a = 2, d = 2 and l = 30 Let the number of terms be n. Then, a + (n - 1)d = 30 ⟹ 2 + (n - 1) x 2 = 30 ⟹ n = 15. ∴Sn = n (a + l) = 15 x (2 + 30) = (15 x 16) = 240. 2 2

Question 5

On multiplying a number by 7, the product is a number each of whose digits is 3. The smallest such number is:

Accepted Answer

By hit and trial, we find that 47619 x 7 = 333333.

Question 6

A number when divided by 296 leaves 75 as remainder. When the same number is divided by 37, the remainder will be:

Accepted Answer

Let x = 296 q + 75 = (37 x 8q + 37 x 2) + 1 = 37 (8q + 2) + 1 Thus, when the number is divided by 37, the remainder is 1.

Question 7

A number when divided successively by 4 and 5 leaves remainders 1 and 4 respectively. When it is successively divided by 5 and 4, then the respective remainders will be

Accepted Answer

4 | x            y = (5 x 1 + 4) = 9
 --------
 5 | y -1         x = (4 x y + 1) = (4 x 9 + 1) = 37
 --------
   | 1 -4

 Now, 37 when divided successively by 5 and 4, we get

 5 | 37
 ---------
 4 | 7 - 2
 ---------
   | 1 - 3
 
 Respective remainders are 2 and 3.

Question 8

(935421 x 625) =?

Accepted Answer

935421 x 625 = 935421 x 54 = 935421 x ❨ 10 ❩ 4 2 = 935421 x 104 = 9354210000 24 16 = 584638125

Question 9

753 x 753 + 247 x 247 - 753 x 247 =? 753 x 753 x 753 + 247 x 247 x 247

Accepted Answer

Given Exp. = (a2 + b2 - ab) = 1 = 1 = 1 (a3 + b3) (a + b) (753 + 247) 1000

Question 10

A number was divided successively in order by 4, 5 and 6. The remainders were respectively 2, 3 and 4. The number is:

Accepted Answer

4 | x            z = 6 x 1 + 4  = 10
 -----------
 5 | y -2         y = 5 x z + 3  = 5 x 10 + 3  = 53
 -----------
 6 | z - 3        x = 4 x y + 2  = 4 x 53 + 2  = 214
 -----------
   | 1 - 4
   
Hence, required number = 214.

Question 11

Which one of the following is not a prime number?

Accepted Answer

91 is divisible by 7. So, it is not a prime number.

Question 12

If the number 91876 * 2 is completely divisible by 8, then the smallest whole number in place of * will be:

Accepted Answer

Then number 6 x 2 must be divisible by 8. ∴ x = 3, as 632 is divisible 8.

Question 13

(12)3 x 64 ÷ 432 =?

Accepted Answer

Given Exp. = (12)3 x 64 = (12)3 x 64 = (12)2 x 62 = (72)2 = 5184 432 12 x 62

Question 14

Which of the following numbers is divisible by each one of 3, 7, 9 and 11?

Accepted Answer

639 is not divisible by 7 2079 is divisible by each of 3, 7, 9, 11.

Question 15

What value should come in place of the question mark(?) in the following question? (?) - 19657 - 33994 = 9999

Accepted Answer

19657         Let x - 53651  = 9999
 33994         Then, x = 9999 + 53651 = 63650
 -----
 53651
 -----

Question 16

If x and y are the two digits of the number 653xy such that this number is divisible by 80, then x + y =?

Accepted Answer

80 = 2 x 5 x 8 Since 653xy is divisible by 2 and 5 both, so y = 0. Now, 653x is divisible by 8, so 13x should be divisible by 8. This happens when x = 6. ∴x + y = (6 + 0) = 6.

Question 17

If the number 481 * 673 is completely divisible by 9, then the smallest whole number in place of * will be:

Accepted Answer

Sum of digits = (4 + 8 + 1 + x + 6 + 7 + 3) = (29 + x), which must be divisible by 9. ∴ x = 7.

Question 18

397 x 397 + 104 x 104 + 2 x 397 x 104 =?

Accepted Answer

Given Exp. = (397)2 + (104)2 + 2 x 397 x 104 = (397 + 104)2 = (501)2 = (500 + 1)2 = (5002) + (1)2 + (2 x 500 x 1) = 250000 + 1 + 1000 = 251001

Question 19

If the number 42573 * is exactly divisible by 72, then the minimum value of * is:

Accepted Answer

72 = 9 x8, where 9 and 8 are co-prime. The minimum value of x for which 73x for which 73x is divisible by 8 is, x = 6. Sum of digits in 425736 = (4 + 2 + 5 + 7 + 3 + 6) = 27, which is divisible by 9. ∴Required value of * is 6.

Question 20

1904 x 1904 =?

Accepted Answer

1904 x 1904 = (1904)2 = (1900 + 4)2 = (1900)2 + (4)2 + (2 x 1900 x 4) = 3610000 + 16 + 15200. = 3625216.

Numbers Questions