### 31st Putnam 1970

Problem A3

A perfect square has length n if its last n digits (in base 10) are the same and non-zero. What is the longest possible length? What is the smallest square achieving this length?

Solution

All squares end in the digit 0, 1, 4, 9, 6, or 5. A square ending in 11, 99, 66, 55 would be congruent to 3, 3, 2, 3 mod 4, but squares are congruent to 0 or 1 mod 4. So for length greater than 1 the square must end in 4. For example, 122 = 144.

A square ending in 4444 would be congruent to 12 mod 16, but squares are congruent to 0, 1, 4 or 9 mod 16. So the maximum length is 2 or 3.

If n2 ends in 4, then n must end in 2 or 8. (100a + 10b + 2)2 = 10000a2 + 1000 (2ab) + 100 (4a + b2) + 10 (4b) + 4. So if this ends in 44, then b = 1 or 6. If b = 1, then 4a + b2 is odd, so the square cannot end in 444. If b = 6, then the square is 1000k + (4a + 38) 100 + 44. This will end in 444 if we take a = 4 or 9. Thus the smallest numbers ending in 2 whose square ends in 444 are 462 (square 213444) and 962 (square 925444).

(100a + 10b + 8)2 = 1000k + 100 (16a2 + b2) + 10 (16b + 6) + 4. So if this ends in 44, then b = 3 or 8. (100a + 38)2 = 1000k + 100 (76a) + 1444, so this ends in 444 if a = 0. This must be the smallest solution.