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If p and q are integers greater than 1 and each is a divisor of the integer Z, which of the following must divide Z^{pq} ? I. p^{q - 1} II. q^{p - 1} III. p + q - 1

Explanation	Calculations	Help
$We translate the fact that "p divides Z" into an equation.$	$Z = p \times m$	Theory & Tactics Method Card TRANS5-E Click to view full details
$We then raise both sides to the power pq to express Z^{pq} in terms of p and m .$	$Z^{pq} = (p \times m)^{pq} = p^{pq} \times m^{pq}$
$We apply "Exponent Rule for Powers" . Since p divides Z, raising Z to the pq power makes it divisible by p raised to the pq power. Because pq is at least q - 1, p raised to q - 1 divides p raised to pq, and so divides Z^{pq} .$	$p^{pq} = p^{q - 1} \times p^{pq - (q - 1)}$	Theory & Tactics Method Card DIV5-A Click to view full details
$We apply "Exponent Rule for Powers" again for q . Since q divides Z, raising Z to the pq power makes it divisible by q raised to the pq power. Because pq is at least p - 1, q raised to p - 1 divides Z^{pq} .$	$q^{pq} = q^{p - 1} \times q^{pq - (p - 1)}$	Theory & Tactics Method Card DIV5-A Click to view full details
$We give a counterexample using p = 2, q = 4, and Z = 4 to illustrate that p + q - 1 need not divide Z^{pq} .$	$p = 2, q = 4, Z = 4 Z^{pq} = 4^{8} = 65536 p + q - 1 = 5 5 does not divide 65536$

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E

If p and q are integers greater than 1 and each is a divisor of the integer Z, which of the following must divide Z^{pq} ? I. p^{q - 1} II. q^{p - 1} III. p + q - 1

Explanation	Calculations	Help
$We translate the fact that "p divides Z" into an equation.$	$Z = p \times m$	Theory & Tactics Method Card TRANS5-E Click to view full details
$We then raise both sides to the power pq to express Z^{pq} in terms of p and m .$	$Z^{pq} = (p \times m)^{pq} = p^{pq} \times m^{pq}$
$We apply "Exponent Rule for Powers" . Since p divides Z, raising Z to the pq power makes it divisible by p raised to the pq power. Because pq is at least q - 1, p raised to q - 1 divides p raised to pq, and so divides Z^{pq} .$	$p^{pq} = p^{q - 1} \times p^{pq - (q - 1)}$	Theory & Tactics Method Card DIV5-A Click to view full details
$We apply "Exponent Rule for Powers" again for q . Since q divides Z, raising Z to the pq power makes it divisible by q raised to the pq power. Because pq is at least p - 1, q raised to p - 1 divides Z^{pq} .$	$q^{pq} = q^{p - 1} \times q^{pq - (p - 1)}$	Theory & Tactics Method Card DIV5-A Click to view full details
$We give a counterexample using p = 2, q = 4, and Z = 4 to illustrate that p + q - 1 need not divide Z^{pq} .$	$p = 2, q = 4, Z = 4 Z^{pq} = 4^{8} = 65536 p + q - 1 = 5 5 does not divide 65536$

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E