



 
Jyll, Working in base ten, you can add, subtract, or multiply using just the units digit  that is, dropping any largervalue 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 2^{7}) or in 2 (like 2^{17}). 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.
 


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