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# Base Number Arithmetic

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If S, H, and E are all distinct non-zero digits less than 5 and the following is true, find the sum of the three values S, H, and E, expressing your answer in base 5.

$$\begin{array}{c@{}c@{}c@{}c} &S&H&E_5\\ &+&H&E_5\\ = &S&E&S_5\\ \end{array}$$

Thanks!

Sep 11, 2020

#1
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This was actually pretty hard! Also, I suggest not to post these types of questions because it suggests something.... Math is not a place for this kind of joking around.

Ok, let's begin.

In base 10, we write expanded notation of a number xyz as 100x+10y+z, or 10^2*x+10^1*y+10^0*z. This works the same in other bases.

Therefore we have 25S+5H+E+5H+E=25S+5E+5S in base 10. Next we combine like terms, getting 2E+10H+25S=30S+5E, and we need to find positive integer solutions to S,H, and E that are less than 10, or digits. Simplifying like terms again, we get 10H=5S+3E. It's pretty obvious what S, E, and H are now. By guess and check, H=2, S=2, and E=0. 220 is SHE in base 5. I don't think this question has an answer though because 220+20 is not 202.

Sep 11, 2020
#2
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S H E_5      = 4 1 2

H E_5       =    12

_______

S E S_5       =4 2 4

Sep 11, 2020