Exercise 6

49. Find the no. of zeros at the end of the product 64 x 625 x 31 x 121 x 12 x 30?
a) 7
b) 5
c)3
d) 4

Explanation:26 x 54 x 31 x 112 x 22 x 31 x 21 x 31 x 51
= 29 x 55 (minimum power is 5)
=5
= 00000

50)Find the no. of prime factor in the expression 12 x 18 x 30 x 40?
a) 13
b) 20
c) 5
d) 31

Explanation:Here The formula is: ap + bq + cr = p + q + r;
22 x 31 x 21 x 32 x 21 x 31 x 51 x 23 x 51
= 27 x 34 x 52
= 7 + 4 + 2
= 13

51)Find the no of divisors in the expression 12 x 18 x 30 x 40?
a)51
b)131
c)120
d)233

Explanation:the formula is: (p + 1) (q + 1) (r + 1)
22 x 31 x 21 x 32 x 21 x 31 x 51 x 23 x 51
=27 x 34 x 52
= (7 + 1) (4 + 1) (2 + 1)
=8 x 5 x 3
=120

44)Which of the following will divide 325 + 3 26 + 327 + 328
a)Only Even Numbers
b)Only Odd Numbers
c)3,4,5,6...
d) Only Prime Numbers

Explanation:

325 + 3 26 + 327 + 328
=>325(1 + 3 + 32 + 33)
=>325(1 + 3 + 9 + 27)
=>325(40)
Divided by = 3, 4, 5, 6, 8, 9....

45) 1/2 + 1/6 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 ?
a) 5/6
b) 6/7
c) 7/8
d) 8/9

Explanation:

1 - 1/2 + 1/2 -1/3 + 1/3 -1/4 + 1/4 - 1/5 + 1/5 -1/6 + 1/6 - 1/7 + 1/7 - 1/8
=>1 - 1/8
=>7/8

46) If both 112 and 32 are factors of the number a x 43 x 6 x 1311, which is the smallest possible value of 'a'?
a) 121
b)3267
c)363
d) 37

Explanation:

4,6,13 are not divisible by 11, Hence a should include 112
now(43 x 6 x 1311) / 32
=>4 x 4 x 4 x 3 x 2 x 1311 / 3 x 3
=>That means a will include 3 also.
Minimum value of 'a' = 112 x 3
=363

47) There is a number consisting of two digits, the digit in the unit place is twice of that in the tens place and if two is subtracted from the sum of the digits then it is equal to 1/6th of the number. The number is:
a) 24
b)36
c)42
d)48

Explanation:

Number = 2x + 10x
=>Sum of the digits: x + 2x - 2 = (2x + 10x) / 6
=>3x - 2 = 12x / 6
=> 18x -12 = 12x
=> 6x = 12
=> x = 2
So, the number is 24

48) A number consists of two digits such that the digits in the ten's place is less by 2 then the digit in the unit place. Three times the number added to 6/7 times the number obtain by reversing the digits equals 108. The sum of digits of the number is:
a) 8
b) 9
c) 6
d) 7

Explanation:

Unit Digit = x
Tens Digit = (x-2)
Number = (x + 10x -20)
On reversing the digits, (x - 2 + 10x)
3(x + 10x - 20) + 6/7(x - 2 + 10x) = 108
=> 3x + 30x - 60 + 6x/7 - 12/7 + 60/7 = 108 x 7
=> 21x + 210x - 420 + 6x - 12 + 60x = 756
=>x = 1188/237
=>x = 4
x - 2 = 4 - 2 = 2
4 + 2 = 6