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# How many zeros in the solution of 65^20-65^16

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Hi :)

how to calculate how many zeros is in the solution of 65^20-65^16?

Jan 25, 2019

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How many zeroes are in the solution of 65^20-65^16

$$\begin{array}{|rcll|} \hline && 65^{20}-65^{16} \\ &=& 65^{16}(65^4-1) \\ &=& 65^{16}(65^2-1)(65^2+1) \\ &=& (5\cdot 13)^{16}(4224)(4226) \quad \text{Decomposition into prime factors} \\ &=& (5\cdot 13)^{16}(2^7\cdot 3 \cdot 11)(2\cdot 2113 ) \quad 10~\text{is the factor of 5 and 2 } \\ &=& 2^8\cdot 5^{16}\ldots \quad \text{The minimum of 16 and 8 is 8 is the number of zeroes in the product. } \\ \hline \end{array}$$

In the solution of $$\mathbf{65^{20}-65^{16}}$$ are $$\mathbf{8}$$ zeroes.

Proof  $$65^{20}-65^{16} = 1\ 812\ 454\ 482\ 098\ 982\ 309\ 886\ 289\ 062\ 500\ 000\ 000$$

Jan 25, 2019
#2
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I can't express how much I appreciate your help!!! Thank you a million^1000

Guest Jan 25, 2019