# How many natural numbers | A

• Feb 16th 2009, 10:21 AM
metlx
How many natural numbers | A
A is the smallest natural number with a feature, that

$10 \times A = perfect square$
$6 \times A = perfect cube$

How many natural numbers | A?
(I don't know how to say | in english :P)

(A) 30
(B) 40
(C) 54
(D) 72
(E) 96
• Feb 16th 2009, 11:10 AM
stapel
Quote:

Originally Posted by metlx
A is the smallest natural number with a feature, that

$10 \times A = perfect square$
$6 \times A = perfect cube$

How many natural numbers | A?

For 10A to be a perfect square, then you must have (2*5)(A) = m^2, for some number m. This number m must include the factors 2 and 5, along with whatever A brings to the table. For 2*5*A to be a square, then there must be a 2 and a 5 as factors of A. So A = 2*5*(something else squared), so m^2 = (2^2)(5^2)(something else squared).

For 6A to be a perfect cube, then you must have (2*3)(A) = n^2, for some number n. This number n must include the factors 2 and 5, along with whatever A brings to the table. For 2*3*A to be a cube, then there must be two 2s and two 3s as factors of A. So A = (2^2)(3^2)(something else cubed), so n^3 = (2^3)(3^3)(something else cubed).

So what factors, at a minimum, must be included in A? (Note that the "something else squared" and "something else cubed" could be just 1^2 and 1^3, respectively.) For instance, since you [i]must have a 5 in A (from the squaring) and 6A must be a cube, how many copies of 5 must be in A?