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# number theory

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Find all positive integers n, less than 17, for which n! is an integral multiple of 49.

Jun 21, 2021

#1
+288
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Imagine writing down the definition of n! in full

1 x 2 x 3 x 4 x ... x n

if it has both 14 in it then it means it's a multiple of 49 because it has one 7 (in 7) and another 7 (in 14) as prime factors.   So an an n must be at least 14.

Jun 21, 2021
#2
+26321
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Find all positive integers n, less than 17, for which n! is an integral multiple of 49.

$$\boxed{\dfrac{n}{7}=2 \\ \mathbf{n}=\mathbf{14} }$$

Jun 22, 2021