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If p and r are integers greater than 1 and each is a factor of the integer N, which of the following must be a factor of N^{p + r} ? I. p^{r} II. (p r)^{2} III. p + r

Explanation	Calculations	Help
$We start by translating "p divides N" .$	$N = p \times k$	Theory & Tactics Method Card TRANS5-E Click to view full details
$We then raise both sides to the power of p + r.$	$N^{p + r} = (p \times k)^{p + r} = p^{p + r} \times k^{p + r}$	Theory & Tactics Method Card DIV5-A Click to view full details
$We note that p + r is at least r, so we get r copies of p in the product and p^{r} divides p raised to the p + r power.$	$p^{p + r} = p^{r} \times p^{p}$	Theory & Tactics Method Card DIV5-A Click to view full details
$Since p and r are both greater than 1, their sum is at least 2, so we get two copies each of p and r when raising to that power and thus two copies of the product p times r.$	$(p r)^{2} = p^{2} \times r^{2}$	Theory & Tactics Method Card DIV5-A Click to view full details
$We test whether p + r must always be a factor by letting p = 2, r = 3, and N = 6.$	$6^{5} = 7776 \frac{7776}{5} = 1555 remainder 1$

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E

If p and r are integers greater than 1 and each is a factor of the integer N, which of the following must be a factor of N^{p + r} ? I. p^{r} II. (p r)^{2} III. p + r

Explanation	Calculations	Help
$We start by translating "p divides N" .$	$N = p \times k$	Theory & Tactics Method Card TRANS5-E Click to view full details
$We then raise both sides to the power of p + r.$	$N^{p + r} = (p \times k)^{p + r} = p^{p + r} \times k^{p + r}$	Theory & Tactics Method Card DIV5-A Click to view full details
$We note that p + r is at least r, so we get r copies of p in the product and p^{r} divides p raised to the p + r power.$	$p^{p + r} = p^{r} \times p^{p}$	Theory & Tactics Method Card DIV5-A Click to view full details
$Since p and r are both greater than 1, their sum is at least 2, so we get two copies each of p and r when raising to that power and thus two copies of the product p times r.$	$(p r)^{2} = p^{2} \times r^{2}$	Theory & Tactics Method Card DIV5-A Click to view full details
$We test whether p + r must always be a factor by letting p = 2, r = 3, and N = 6.$	$6^{5} = 7776 \frac{7776}{5} = 1555 remainder 1$

Scroll horizontally to view all columns

E