Problem 8 · 1986 AJHSME
Medium
Number Theory
last-digitguess-and-check
Show answer
Answer: E — 8.
Show hints
Hint 1 of 3
Don't try every digit 0–9. The very last digit of the answer (the 6 in 6396) is decided only by the last digits being multiplied — so look at the ones column first.
Still stuck? Show hint 2 →
Hint 2 of 3
The ones digit of B2 is 2 and of 7B is B, so the product's ones digit comes from 2 × B. For what B does 2 × B end in 6? That narrows ten guesses down to two.
Still stuck? Show hint 3 →
Hint 3 of 3
With only two candidates left, a single quick multiplication tells you which one is right.
Show solution
Approach: ones-digit filter, then verify the survivor
- Attack the ones column first — it's independent of all the carrying. The ones digits being multiplied are 2 (from B2) and B (from 7B), so 2 × B must end in the product's last digit, 6.
- 2 × B ends in 6 only when B = 3 (gives 6) or B = 8 (gives 16). That's the whole point: a 10-way search collapses to 2.
- Check the survivors: 32 × 73 = 2336 (too small), but 82 × 78 = 6396 ✓. So B = 8.
- Why this transfers: in any 'find the hidden digit' puzzle, the ones digit of a product depends only on the ones digits of the factors — start there to slash your candidates before doing real arithmetic.
Mark:
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