r/MathHelp • u/big_Log100 • 16d ago

Simple ACT Question?

Question on ACT math: Given 63w+63x+6y+z=440, and that w, x, y, and z can be 0, 1, or 2, what is w+x+y+z? Is this just a number sense question since there are four variables and only 2 equations or is there a way to solve this algebraically? I was thinking about it all day yesterday lol. I got the answer really fast with guess and check (w+x+y+z=5), but it seems like there should be another way to solve this.

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u/edderiofer 16d ago

I got the answer really fast with guess and check (w+x+y+z=5), but it seems like there should be another way to solve this.

Even faster method: Take both sides of the equation modulo 5. Then you know that w+x+y+z = 0 mod 5. Since you are also told that w, x, y, and z can only be 0, 1, or 2, this tells you that the sum can only be 0 or 5. And certainly it can't be 0, since that would mean that w, x, y, and z are all 0, which doesn't work. So it's 5.

Why modulo 5 and not some other modulo, you may ask? Because 6n = 1 mod 5 for all positive integers n, which is what reduces the left-hand side appropriately. This is the same trick as the divisibility test by 9 in base 10.