phoenixgmat
I would appreciate some help with:
If the prime numbers p and t are the only prime factors of the integer m, is m a multiple of p²t?
1) m has more than 9 positive factors.
2) m is a multiple of p³
some explanations to both statements would be great!
thx a lot
We are told that \(p\) and \(t\) are the ONLY prime factors of m. It could be expressed as \(m=p^x*t^y\), where \(x\) and \(y\) are integers \(\geq{1}\).
Question: is \(m\) a multiple of \(p^2*t\). We already know that \(p\) and \(t\) are the factors of \(m\), so basically question asks whether the power of \(p\), in our prime factorization denoted as \(x\), more than or equal to 2: so is \(x\geq{2}\).
(1) m has more than 9 positive factors:
Formula for counting the number of distinct factors of integer \(x\) expressed by prime factorization as: \(n=a^x*b^y*c^z\), is
\((x+1)(y+1)(z+1)\). This also includes the factors 1 and \(n\) itself.
We are told that \((x+1)(y+1)>9\) (as we know that \(m\) is expressed as \(m=p^x*t^y\))
But it's not sufficient to determine whether \(x\geq{2}\). (\(x\) can be 1 and \(y\geq{4}\) and we would have their product \(>9\), e.g. \((1+1)(4+1)=10\).) Not sufficient.
(2) m is a multiple of p^3
This statement clearly gives us the value of power of \(p\), which is 3, \(x=3>2\). So \(m\) is a multiple of \(p^2t\). Sufficient.
Answer: B.
I could answer the question in 1.5 min. But I have never known the formula for counting the number of distinct factors of a integer. Thanks a ton, Bunuel