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# Prime Factorization

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In the prime factorization of $109!$, what is the exponent of $3$? (Reminder: The number $n!$ is the product of the integers from 1 to $n$. For example, \$5!=$$5\cdot 4\cdot3\cdot2\cdot 1= 120.)$$

Apr 6, 2020

#1
+24983
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In the prime factorization of $$109!$$, what is the exponent of $$3$$?

(Reminder: The number $$n!$$ is the product of the integers from $$1$$ to $$n$$.

For example, $$5!=5\cdot 4\cdot3\cdot2\cdot 1= 120$$.)

$$\begin{array}{|rcll|} \hline \dfrac{109}{3^1} &=& \lfloor 36.3333333333 \rfloor \\ &=& 36 \\ \hline \dfrac{109}{3^2} &=& \lfloor 12.1111111111 \rfloor \\ &=& 12 \\ \hline \dfrac{109}{3^3} &=& \lfloor 4.03703703704 \rfloor \\ &=& 4 \\ \hline \dfrac{109}{3^4} &=& \lfloor 1.34567901235 \rfloor \\ &=& 1 \\ \hline \dfrac{109}{3^5} &=& \lfloor 0.44855967078 \rfloor \\ &=& 0 \\ \ldots \\ &=& 0 \\ \hline \text{sum} && 36+12+4+1= \mathbf{53} \\ \hline \end{array}$$

$$\mathbf{109!} = 2^{104}×\mathbf{3^{\color{red}53}}×5^{25}×7^{17}×11^9×13^8×17^6×19^5×23^4×\cdots$$

Apr 6, 2020
#2
+392
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thank u so much

Apr 6, 2020