The exponent of $11$ in the prime factorization of $ 300!$ is___.

Question

The exponent of $11$ in the prime factorization of $ 300!$ is

1. $27$
2. $28$
3. $29$
4. $30$

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My attempt:

According to Exponent of $p$ in the prime factorization of $n!$

$\left\lfloor\dfrac{300}{11}\right\rfloor+\left\lfloor\dfrac{300}{11^2}\right\rfloor=27+2=29$

Can you explain in alternative/formal way? Please.

Answer 1

See the terms having factor $11$ are $11,22,33,..121,...220,..297$ so excluding $121,242$ we have $25$ numbers which give only one $11$ and $121,242$ gives two $11$ so total exponent is $11^{25}.11^4=11^{29}$