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In the prime factorization $37! = 2^{α_{2}} 3^{α_{3}} 5^{α_{5}} - - - 37^{α_{37}}$ then the ratio $α_{3} : α_{5} =$

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$17 : 8$

$3 : 5$

$5 : 3$

$8 : 21$

answer is B.

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Detailed Solution

$α_{3} = [\frac{37}{3}] + [\frac{37}{3^{2}}] + [\frac{37}{3^{3}}] + [\frac{37}{3^{4}}] + ......$
=12+4+1+0
=17
$α_{5} = [\frac{37}{5}] + [\frac{37}{5^{2}}] + [\frac{37}{5^{3}}] + ......$
=7+1+0
=8
$α_{3} : α_{5} = 17 : 8$

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