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What is the sum of the digits of the number $(5^{2015})(2^{2018})$

So I am guessing, I have to find out the product of $(5^{2015})(2^{2018})$ and add each digit of the product.

The question is how do I find the product of $(5^{2015})(2^{2018})$. Both numbers don't share the same base or exponents, so none of the laws of exponents (that I know of) will help me. Unless this problem is meant to be clever and have another way.

Any leads?

dmtri
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Caddy Heron
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    Notice that $5^x\cdot2^x=10^x$. Does this help? – Tim Thayer Sep 28 '15 at 03:38
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    Note that $5^{2015} \times 2^{2018}$ = $(2\times 5)^{2015} \times 2^3 $. – stochasticboy321 Sep 28 '15 at 03:38
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    @dmtri: This is what I meant with "avoid gaming the system". If you feel the need to edit old questions, find the ones that really need to be edited. Not just capitalisations of "i" or adding $ around p. The older the thread, the more substantial the edit should be. – Asaf Karagila Jan 09 '20 at 10:02
  • @Asaf Karagila, thanks for making the point. I will follow. – dmtri Jan 09 '20 at 11:59

2 Answers2

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$$(5^{2015})(2^{2018}) = (5^{2015})(2^{2015})(2^3) = (5\cdot 2)^{2015}2^3$$ $$= 10^{2015}2^3 = 8\cdot 10^{2015}$$ Notice that this is just the digit $8$ followed by $2015$ zeroes, so the sum is just $8$

Brevan Ellefsen
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Hint: $(5)^{2015}(2)^{2015} = (10)^{2015}$.

$(2)^{2018} = 2^{2015} \cdot 2^3$

Deepak
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