Question

How do you find a and b?

i) a is such that has 12 divisors
ii) b has 18 divisors and
iii) gcd (a,b)=45

Answer 1

`a=90`
`b=171`

Explanation:

`a=45x_1`

`b=45x_2`

Divisors of `45 -> 1xx45, color(white)("d")3xx15,color(white)("d")5xx9` count of 6

Given that `a->12` divisors then `2xx45->2xx"6 divisors"`

Given that `b->18` divisors then `3xx45->3xx"6 divisors"`

`3xx45= 171`
`2xx45=90`

Answer 2

Another approach.

Explanation:

Given a number `N = prod_(k=1)^n p_k^(alpha_k)`

with `p_k` being the primes and `alpha_k` their multiplicity we have

`prod_(k=1)^n (alpha_k+1)` divisors for `N`

Assuming now that `a` and `b` are composed by the same prime factors `3` and `5` we have

`a = 3^2 xx 5^(alpha_1)`
`b = 3^(alpha_2) xx 5`

The number of factors for `a` and `b` are

`{(a->(2+1)(alpha_1+1)= 12),(b->(alpha_2+1)(1+1)=18):}`

now solving for `alpha_1, alpha_2` we have

`alpha_1 = 3` and `alpha_2 = 8`

then

`a = 3^2 xx 5^3` and
`b = 3^8 xx 5`

`a` factors are

`{1, 3, 5, 9, 15, 25, 45, 75, 125, 225, 375, 1125}`

`b` factors are

`{1, 3, 5, 9, 15, 27, 45, 81, 135, 243, 405, 729, 1215, 2187, 3645, 6561, 10935, 32805}`