Jyll,

Working in base ten, you can add, subtract, or multiply using just the units digit - that is, dropping any larger-value digits - and the results will be well defined. Thus we know that XXXX4 + XXX3 ends in 7, no matter what the X's hide; and XXXX4 x XXXX3 ends in 2.

We can do divisions only if the number we are dividing by is coprime to 10 - that is, is 1,3,7 or 9. So we know XXXX7 / XXX3, if it is a whole number, ends in 9; but XXXX2 / XX2 could end in 1 or 6.

Not all powers can be worked out on the basis of last digits only. If we raise a number ending in 0,1,5, or 6 to any power the answer is always the same; for instance, XXXX6^XXXXX must end in 6. If we raise a number ending in 4 or 9 to a power, we only need to know if the power is odd or even, which we can tell from a last digit; so we know XXXX4^XXX8 ends in 6, like all even powers of 4. But powers of 2,3,7, and 8 cycle with period 4, so XXX2 ^ XXX7 could end in 8 (like 2⁷) or in 2 (like 2¹⁷).

And that, what is the units digit of the first 1000 prime numbers?

I don't know any way to find that which is easier than finding the first 1000 primes completely. Of course, 2 and 5 will each appear once near the beginning and no more; all the rest will end in 1,3,7 or 9.


Good Hunting!
RD