What is the units digit of 31467^32 × 97645^23 × (32168^5 + 8652)^479?

For ANSWER let's see only the units digit..
\(7^{32}*5^{23}*(8^5+2)^{479}\)
... This means \(7^4*5*(8^1+2)^{480-1}\)......
after every 4th power, the units digit repeat
for units digit 0, 1, 5 and 6... it remains the same for every positive integer power
for 2,3,7,8 it repeats after every 4th.. so 2^401 will be same as 2^(4*50+1) or 2^1
4 and 9 repeat after every second..
so \(7^4*5\) will be 5 as ODD *5 will have units digit 5
but \((8^1+2)^{479} = 10^{479}\) will always have 0 as units digit..
so overall also it will be 0

A

Could not have put it in any better way!!

chetan2u , I'm not clear about why you wrote this part: \((8^1+2)^{480-1}\). (480/4) is evenly divisible, maybe the (-1) is a negative remainder (and I will never understand negative remainders fully) but that doesn't seem to explain the exponent manipulation, either. . .

Was it to show remainder in case \((8^1+2)\) summed to something other than zero? If that is the reason, I still don't follow.

You can't distribute the exponent, so cyclicity of four for addends 8 and 2 can't be the reason.

And cyclicity per se doesn't seem to be the reason: 2, 3, 7, and 8 have cyclicity of four (where 480 is divisible by 4) - but 0, 1, 4, 5, 6, and 9 do not have cyclicity of four. That 8 might have had units digit 2, e.g. Then units digit is 4, and I can't see how \(4^{480-1}\) helps.

The (-1) part isn't clear, either. I suppose I know that, in the number whose cyclicity of powers is four, I can take the \(n^4\) term and "move backwards" by one power in the cycle to get the units digit, but 479/4 leaves remainder 3, which is the same thing as "moving one place backward" in the cycle. It seems harder to see, from a divisibility perspective, that (-1) power in cycle = (+3) power in cycle.

I looked only at \(8^5\), where 5/4 = remainder 1, units digit of first term is same as \(8^1\) = 8. Last digit of addend in parentheses is 2. (8 + 2) = 0.

And as you point out, zero to any power (except zero) is 0. So I just wrote (first expression) * 0.

Sorry if this confusion arises because I am missing something obvious. I suspect what you wrote is important for problems different from this one. Please explain?