a)31
b) 61
c) 71
d) 91
91 is divisible by 7. So, it is not a prime number.
(2) (112 x 54) =?
a) 65000
b) 70000
c) 72000
d) 75000
112 x 25 x 25 = 70000
(3) It is being given that (232 + 1) is completely divisible by a whole number.
a) (2^16 + 1)
b) (2^64 + 1)
c) (2^54 + 1)
d) (2^96 + 1)
Let 2^32 = x. Then, (2^32 + 1) = (x + 1).
Let (x + 1) be completely divisible by the natural number N. Then,
(2^96 + 1) = [(2^32)^3 + 1] = (x^3 + 1) = (x + 1)(x^2 - x + 1), which is completely divisible by N, since (x + 1) is divisible by N.
(4) What least number must be added to 1056, so that the sum is completely divisible by 23?
a) 1
b) 2
c) 6
d) 3
When 1056 is divided by 23, Remainder is 21. So, 2 must be added to get completely divided by 23.
(5) 1397 x 1397 = ?
a) 1941609.
b) 1951709.
c) 1961609.
d) 1951609.
| 1397 x 1397 | = (1397)^2 |
| = (1400 - 3)^2 | |
| = (1400)^2 + (3)^2 - (2 x 1400 x 3) | |
| = 1960000 + 9 - 8400 | |
| = 1960009 - 8400 | |
| |
= 1951609. |
(6) What is the unit digit in {(6374)^1793 x (625)^317 x (341^491)}?
a) 0
b) 5
c) 2
d) 1
Unit digit in (6374)^1793 = Unit digit in (4)^1793
= Unit digit in [(42)^896 x 4]
= Unit digit in (6 x 4) = 4
Unit digit in (625)^317 = Unit digit in (5)^317 = 5
Unit digit in (341)^491 = Unit digit in (1)^491 = 1
Required digit = Unit digit in (4 x 5 x 1) = 0.
(7) The sum of first five prime numbers is:
a) 28
b) 18
c) 22
d) 39
Required sum = (2 + 3 + 5 + 7 + 11) = 28.
Note: 1 is not a prime number.
(8) The difference of two numbers is 1365. On dividing the larger number by the smaller, we get 6 as quotient and the 15 as remainder. What is the smaller number?
a) 260
b) 280
c) 270
d) 290
Let the smaller number be x. Then larger number = (x + 1365).
x + 1365 = 6x + 15
5x = 1350 x = 270Smaller number = 270.(9) If the number 517*324 is completely divisible by 3, then the smallest whole number in the place of * will be:
a) 0
b) 1
c) 2
d) 3
Sum of digits = (5 + 1 + 7 + x + 3 + 2 + 4) = (22 + x), which must be divisible by 3.
x = 2.
(10 )Which one of the following numbers is exactly divisible by 11?
a) 235641
b) 245642
c) 315624
d) 415624
(4 + 5 + 2) - (1 + 6 + 3) = 1, not divisible by 11.
(2 + 6 + 4) - (4 + 5 + 2) = 1, not divisible by 11.
(4 + 6 + 1) - (2 + 5 + 3) = 1, not divisible by 11.
(4 + 6 + 1) - (2 + 5 + 4) = 0, So, 415624 is divisible by 11.
