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
First 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
22-7=16;
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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