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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^{s t} ? I. s^{t} II. (s t)^{2} III. s + t

Explanation	Calculations	Help
$Express n in terms of s since s divides n .$	$n = s \times k$
$Raise both sides to the power s t to relate n^{s t} to s^{s t} .$	$n^{s t} = (s \times k)^{s t} = s^{s t} \times k^{s t}$
$Since s t \geq t, the factor s^{t} appears in s^{s t}, so s^{t} divides n^{s t} .$	$s^{s t} = s^{t} \times s^{s t - t}$
$Because s > 1 and t > 1, we have s t \geq 2 . Hence both s^{2} and t^{2} divide n^{s t}, and their product (s t)^{2} divides n^{s t} .$	$(s t)^{2} = s^{2} \times t^{2}$
$Provide a counterexample to show s + t need not divide n^{s t} .$	$s = 2, t = 3, n = 6 6^{6} will end in 6, so s + t = 5 does not divide 6^{6} .$

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I and II (E)

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^{s t} ? I. s^{t} II. (s t)^{2} III. s + t

Explanation	Calculations	Help
$Express n in terms of s since s divides n .$	$n = s \times k$
$Raise both sides to the power s t to relate n^{s t} to s^{s t} .$	$n^{s t} = (s \times k)^{s t} = s^{s t} \times k^{s t}$
$Since s t \geq t, the factor s^{t} appears in s^{s t}, so s^{t} divides n^{s t} .$	$s^{s t} = s^{t} \times s^{s t - t}$
$Because s > 1 and t > 1, we have s t \geq 2 . Hence both s^{2} and t^{2} divide n^{s t}, and their product (s t)^{2} divides n^{s t} .$	$(s t)^{2} = s^{2} \times t^{2}$
$Provide a counterexample to show s + t need not divide n^{s t} .$	$s = 2, t = 3, n = 6 6^{6} will end in 6, so s + t = 5 does not divide 6^{6} .$

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I and II (E)