Problem 4 · 1990 AJHSME
Medium
Number Theory
units-digit-of-squares
Which of the following could not be the units digit (one's digit) of the square of a whole number?
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Answer: E — 8.
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Hint 1 of 2
When you multiply two numbers, the last digit of the answer depends only on the last digits you multiplied. So a square's last digit only depends on the number's last digit — there are just 10 endings to try (0 through 9).
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Hint 2 of 2
This is the units-digit trick: to find how a number *ends*, ignore everything but the ones digits. Square 0,1,2,…,9 and collect which last digits actually appear.
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Approach: the last digit of a square depends only on the last digit squared
- Key idea: the ones digit of n×n is decided entirely by the ones digit of n. So square only 0–9 and watch the endings: 0→0, 1→1, 2→4, 3→9, 4→6, 5→5, 6→6, 7→9, 8→4, 9→1.
- The only endings that ever show up are 0, 1, 4, 5, 6, 9. Notice they come in mirror pairs (1&9, 2&8, 3&7, 4&6 give the same ending), which is why so few appear.
- 8 is not on the list, so no whole number squared can end in 8.
- *Worth keeping:* a perfect square can only end in 0, 1, 4, 5, 6, or 9 — if a number ends in 2, 3, 7, or 8 it is instantly not a square.
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