Question

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

Answer 1

E ( 1 and 2) only

Explanation:

s is an integer and a factor of n and is greater than 1 and t is also an integer and a factor of and equally greater than 1

s^t is a factor n^(st)

let s = 3 and t = 4 and n = 12

then n^(3×4) = 12 ^12= (3 × 4) ^(3×4) = 3³ × 4⁴ or 3⁴ × 4³

(st)² = (3×4)² = (12)² which is a factor of 12¹²

3^4

s + t = 3+4 = 7 and 7 is not a factor 12¹²

let say s = 2, and t = 3

2 + 3 = 5 and 5 is not a factor of 6⁶

Answer 2

E) 1 and 2

Explanation:

We are given that there are two integers (s and t) and they are factors of another integer (n). For example if s = 3 and t = 2, we can have n = 6.

Thus:

n^(st) = 6^(2*3) = 6^6 = (2^6)(3^6)

For the first condition: s^t = 3^2 is a factor of (2^6)(3^6)

For the second condition: (st)^2 = (3*2)^2 = 6^2 is a factor of 6^6

For the third condition: s+t = 3+2 = 5 is not a factor of 6^6 or (2^6)(3^6)

Therefore, only 1 and 2 are factors of n^(st)