- The number of prime factors of (3 x 5)12 (2 x 7)10 (10)25 is:

A): 47

B): 60

C): 72

D): None of these

Correct Answer : D

Explanation: The equation can be facorize as 3*5*3*2*2*2*7*2*5*2*5*5*5 or 2^5*3^2*5^5*7^1 total no of prime factor =(5+1)*(5+1)*(2+1)*(1+1)=216

- What least value must be assigned to * so that the number 63576*2 is divisible by 8?

A): 1

B): 2

C): 3

D): 4

Correct Answer :C

Explanation: The test for divisibility by 8 is that the last 3 digits of the number in question have to be divisible by 8.

So, 6*2 has to be divisibile by 8.

I know 512 is divisible by 8.

Also 592 is divisible by 8.

So, 632 is divisible by 8.

So * is 3.

- The smallest number, which is a perfect square and contains 7936 as a factor is:

A): 251664

B): 231564

C): 246016

D): 346016

Correct Answer :C

Explanation:

7936 => 2^2 * 2^2 * 2^2 * 2^2 * 31^1

To make it as a perfect square, we have to multiply 7936 with 31…

Hence the reqd no. is 7936*31 = 246016

- In a division problem, the divisor is twenty times the quotient and five times the remainder. If remainder is 16, the number will be:

A): 3360

B): 336

C): 1616

D): 20516

Correct Answer :B

Explanation:

Divisor = (5 x 46) = 230

10 x Quotient = 230 = | 230 | 23 |

10 |

Dividend = (Divisor x Quotient) + Remainder

= (230 x 23) + 46

= 5290 + 46

= 5336.

- If a number is exactly divisible by 85, then what will be the remainder when the same number is divided by 17?

A): 3

B): 1

C): 4

D): 0

Correct Answer : D

Explanation: number=divisor*quotient+remainder

so 17*5+0;

remainder is 0;

divisor is 17;

quotient is 5;

- The least perfect square number which is exactly divisible by 3, 4, 7, 10 and 12 is:

A): 8100

B): 17600

C): 44100

D): None of these

Correct Answer : C

- (xn+yn) is divisible by (x-y):

A): for all values of n

B): only for even values of n

C): only for odd values of n

D): for no values of n

Correct Answer : D

Explanation: for no values of n

- P is an integer. P is greater than 883.If P -7 is a multiple of 11, then the largest number that will always divide (P+4)(P+15) is

A)242

B)343

C)321

D)none

Answer:A

Explanation: p-7= 11*a (as it is multiple of 11)

p=11*(a+7)

so (p+4)(p+15)= (11a+7+4)(11a+7+15);

= (11a+11)(11a+22);

=11*11(a+1)(a+2);

=121*2

=242

- The greatest number that will divide 63, 138 and 228 so as to leave the same remainder in each case:

A): 15

B): 20

C): 35

D): 40

Correct Answer :A

Explanation: The greatest number = H.C.F of (138-63), (228-138), (228-63)

H.C.F of 75, 90, 165 = 15.

15 is the greatest number.

- Find the largest number, smaller than the smallest four-digit number, which when divided by 4,5,6and 7 leaves a remainder 2 in each case:

A): 422

B): 842

C): 12723

D): None of these

Correct Answer : B

Explanation: Take LCM of 4,5,6,7. It is 420

BUt the no must leave remainder 2 in each case, so the no is of the form: 420k + 2.

The smallest 4-digit no is 1000. So keeping k=0,1,2,3….

We get that the largest no smaller than the smallest 4 -digit no is 842

- What is the highest power of 5 that divides 90 x 80 x 70 x 60 x 50 x 40 x 30 x 20 x 10?

A): 10

B): 12

C): 14

D): None of these

Correct Answer : A

Explanation: Take LCM of Each Number

90/5=5*2*3*3——————>here we will get one 5

80/5=5*2*2*2*2—————>here we will get one 5

70/5=5*2*7————–___—->here we will get one 5

60/5=5*2*2*3——————>here we will get one 5

50/5=5*5*2___——————>here we will get Two 5^2

40/5=5*2*2*2——————>here we will get one 5

30/5=5*2*3———————>here we will get one 5

20/5=5*2*2———————>here we will get one 5

10/5=5*2————————>here we will get one 5

Here we will get one 5 in each number instead of 50(5*5*2)

So answer is 5^10

- If a and b are natural numbers and a-b is divisible by 3, then a3-b3 is divisible by:

A): 3 but not by 9

B): 9

C): 6

D): 27

Correct Answer : B

Explanation: If a − b is divisible by 3, then a − b = 3k, for some integer k

(a − b)² = (3k)²

a² − 2ab + b² = 9k²

a³ − b³ = (a−b) (a² + ab + b²)

= (a−b) (a² − 2ab + b² + 3ab)

= 3k (9k + 3ab)

= 3k * 3 (3k + ab)

= 9 k(3k+ab)

Since k(3k+ab) is an integer, then 9k(3k+ab) is divisible by 9

- What is the greatest positive power of 5 that divides 30! exactly?

A): 5

B): 6

C): 7

D): 8

Correct Answer : C

Explanation: The question is, how many powers of 5 are in the factors of 30! (that’s 30factorial, for those above)…

Only the numbers 5, 10, 15, 20, 25, and 30 have divisors of 5. And 25 is divisible by 5^2.

So the answer is 5*5*5*5*(5^2)*5 = 5^7.

- What is the smallest four-digit number which when divided by 6, leaves a remainder of 5 and when divided by 5 leaves a remainder of 3?

A): 1043

B): 1073

C): 1103

D): None of these

Correct Answer : D

Explanation: remainder when m is divided by 5 = 2

Smallest m is 2.

Hence, N = 1001 + 6 * 2 = 1013.

- P is an integer. P>883. If P-7 is a multiple of 11, then the largest number that will always divide (P+4) (P+15) is:

A): 11

B): 121

C): 242

D): None of these

Correct Answer : C

Explanation: Given P is an integer>883.

P-7 is a multiple of 11=>there exist a positive integer a such that

P-7=11 a=>P=11 a+7

(P+4)(P+15)=(11 a+7+4)(11 a+7+15)

=(11 a+11)(11 a+22)

=121(a+1)(a+2)

As a is a positive integer therefore (a+1)(a+2) is divisible by 2.Hence (P+4)(P+15) is divisible by 121*2=242

- Let C be a positive integer such that C + 7 is divisible by 5. The smallest positive integer n (>2) such that C + n2 is divisible by 5 is:

A): 4

B): 5

C): 3

D): Does not exist

Correct Answer : D

Explanation: c + n^2 is divisible by 5 if and only if c and n^2 are both divisible by 5.

But, if c is divisible by 5 then c + 5 will not be divisible by 5.

So, option(d ) is correct.

- Four bells begin to toll together and then each one at intervals of 6 s, 7 s, 8 s and 9 s respectively.
The number of times they will toll together in the next 2 hr is:

A): 14 times

B): 15 times

C): 13 times

D): 11 times

Correct Answer : A

Explanation: first we to find the L.C.M. of 6, 7, 8 and 9.

Prime factorization of 6 = 2*3

Prime factorization of 7 = 7

Prime factorization of 8 = 2*2*2

Prime factorization of 9 = 3*3

L.C.M. = 2*2*2*3*3*7

= 504

The L.C.M. of 6 seconds, 7 seconds, 8 seconds and 9 seconds is 504

seconds.

Now, 1 hour = 3600 seconds

So, 2 hours = 3600*2 = 7200 seconds

The number of times the four bells will toll together in the next 2 hour

= 7200/504

= 14.28 or 14 times

They will toll together 14 times in the next 2 hours

- On dividing a number by 999,the quotient is 366 and the remainder is 103.The number is:

A): 364724

B): 365387

C): 365737

D): 366757

Correct Answer : C

Explanation: Number (Dividend) = Divisor * quotient + remainder.

Number = 999 * 377 + 105 = 3767

- If 522x is a three digit number with as a digit x . If the number is divisible by 6, What is the value of the digit x is?

A)1

B)2

C)3

D)4

e)6

Answer:e

Explanation: If a number is Divisiable by 6 , it must be divisible by both 2 and 3

In 522x, to this number be divisible by 2, the value of x must be even. So it can be 2,4 or 6 from given options

552x is divisible by 3, If sum of its digits is a multiple of 3.

5+5+2+x =12+x ,

If put x =2 , 12+2=14 not a multiple of 3

If put x =4 , 12+6=18 is a multiple of 3

If put x =6 , 12+2=14 not a multiple of 3

The value of x is 6.

Number System Sample Question with Solutions

- P is an integer. P is greater than 883.If P -7 is a multiple of 11, then the largest number that will always divide (P+4)(P+15) is

A)242

B)340

C)245

D)178

Answer:A

Explanation: p-7= 11*a (as it is multiple of 11)

p=11*(a+7)

so (p+4)(p+15)= (11a+7+4)(11a+7+15);

= (11a+11)(11a+22);

=11*11(a+1)(a+2);

=121*2

=242