aniketm.87@gmail.com wrote:
if s and t are integers greater than 1 and each is a factor of the integer n, which of the following must be a factor of n^st?
1) s^t
2) (st)^2
3) s + t
A) None
B) 1 only
C) 2 only
D) 3 only
E) 1 and 2
If s=ax and b=bx, where a, b and x are integers, and as s and t are factors of n, we can write n=LCM(st)*y=abxy
\(n^{st}\) will have factors: \(x^{st}, \ \ s^{st}, \ \ and \ \ t^{st}\)
lets see the choices..
\(1) s^t\)
\(s^{st}\) is a factor, so \(s^t\) and \(s^s\) will also be a factor.. YES
\(2) (st)^2\)
\((s)^{st}\) and \((t)^{st}\) are factors and we know that both s and t are >1. Thus least value of both s and t is 2.
So, the min value of \(n^{st}\) will be \(n^4=(abxy)^4=a^4*b^4*x^4*y^4=a^2*(ax)^2*b^2*(bx)^2*b^2*y^4=a^2*s^2*t^2*y^4=a^2*(st)^2*y^4\).
Thus, \((st)^2\) will always be a factor of \((n)^4\) or \(n^{st}\).. YES
\(3) s + t\)
we do not know what are the factors of s and t...
Say y=5, x=1, a=2, and b=3, that is s=2 and t=3, then n=5*2*3=30. 2+3 or 5 is a factor of \(30^{2*3}\)
But say y=5, x=1, a=2, and b=3, that is s=2 and t=3, then n=6*2*3=36. 2+3 or 5 is not a factor of \(36^{2*3}\)
NOT necessary
Only (1) and (2) must be true.
E
Responding to a PM
Adding my take on substituting some values for s and t. Always be careful on taking values in MUST be true questions. You may have just taken some value that fits in for the values taken but may not fit in for other values.
For example: if you take s=2 and t=6, then n=LCM (2,6)*y=6y
Option III. s+t=2+6=8, which will be a factor of (6y)^12=2^12*3^12*y^12.
Or
If you take s=t, option III will be true irrespective of values of s and t.
thank you for your reply. for "\(s^{st}\) is a factor, so \(s^t\) and \(s^s\) will also be a factor.. YES"
wouldn't \(s^{st}\) factor out to \((s^{t})^s\), not \(s^t\) * \(s^s\)?