Amateur wrote:

Vips0000 wrote:

kingb wrote:

If a natural number ‘p’ has 8 factors, then which of the following cannot be the difference between the number of factors of p3 and p

a. 14

b. 30

c. 32

d. 56

e. None of these

p has 8 factors. That gives following possiblities:

a) p has one prime factor with power 7

=> p^3 will have 22 factors. Diffrerence with number of factors of p = 22-8 = 14: A is ok

b) p has two prime factors with powers 3 and 1

=> p^3 will have 40 factors. Difference with number of facors of p =40-8 = 32 : C is ok

c) p has three prime factors with powers 1 each

=> p^3 will have 64 factors. Difference with number of factors of p=64-8 = 56 : D is ok

There are no other possiblities. Hence remaining answer choice B is not possible.

Ans B it is!

i didnot understand a bit of your explanation.... can you please explain it? thank you

Finding the Number of Factors of an IntegerFirst make prime factorization of an integer

n=a^p*b^q*c^r, where

a,

b, and

c are prime factors of

n and

p,

q, and

r are their powers.

The number of factors of

n will be expressed by the formula

(p+1)(q+1)(r+1).

NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450:

450=2^1*3^2*5^2Total number of factors of 450 including 1 and 450 itself is

(1+1)*(2+1)*(2+1)=2*3*3=18 factors.

BACK OT THE ORIGINAL QUESTION:

We are told that

p has 8 factors: 8=2*4=2*2*2.

If

p has only one prime in its prime factorization, say

a, then

p=a^7 --> number of factors of

p is

(7+1)=8 -->

p^3=(a^7)^3=a^{21} in this case would have

(21+1)=22 factors --> the difference is

22-8=14;

If

p has two primes in its prime factorization, say

a and

b, then:

p=a^3*b --> number of factors of

p is

(3+1)(1+1)=8 -->

p^3=a^9*b^3 in this case would have

(9+1)(3+1)=40 factors --> the difference is

40-8=32;

If

p has three primes in its prime factorization, say

a,

b and

c, then:

p=a*b*c --> number of factors of

p is

(1+1)(1+1)(1+1)=8 -->

p^3=a^3*b^3*c^3 in this case would have

(3+1)(3+1)(3+1)=64 factors --> the difference is

64-8=56.

p cannot have more than 3 factors, since the least number of factors a number with four primes can have is 16>8 (for example if

p=abcd, then number of factors of

p is

(1+1)(1+1)(1+1)(1+1)=16).

Answer: B.

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