Can you show us what you have tried?
The square of a number is just the number times itself. For example, the square of 4 is 16
A.Calculate the squares of 1,2,...,9 and many other whole numbers, including 17, 34, 61, 82, 99, 123, 255, 386, and 728. Record the ones digits in each case. What do you notice? Do any of your squares have a ones digit of 7, for example? are any other digits missing from the ones digits of squares
B.Which digits can never occur as the ones digit of a square of a whole number? Explain why some digits cannot occur as the ones digit of a square of a whole number
C.Based on what you've discovered, could the number 139,787,847,234,329,483 be the square of a whole number? why or why not?
I'm not used to this kind of problem, and I may have missed something, but here's what I think.
Suppose we have a number like 789. The decimal notation means 7*100 + 8*10 + 9.
Any integer of any size can be represented like this:
10*x + n
... where x and n are integers (possibly zero). Now let's square that
(10*x + n)^2 = 100*x^2 + 20*x*n + n^2 = (10*x^2 + 2*x*n)*10 + n^2
The right term, in decimal notation, always ends in zero. This means the unit digit must always come from n^2, and n is an integer between 0 and 9. This means n^2 can be 0, 1, 4, 9, 16, 25, 36, 49, 64, 81 or 100. The unit digit of those numbers can be 0, 1, 4, 9, 6, 5, or 9 or in order 1, 4, 5, 6 or 9. This means the last digit of a square can be one of those five but never 2, 3, 7 or 8.
This strikes me as a mere curiosity, nothing profound in it. I suppose something similar would happen in, say, base 8 or base 12, but is there a pattern that cuts across bases? Probably not.