Problem 23 · 2014 Math Kangaroo
Stretch
Number Theory
factor-pairscasework
Mia writes three single-digit numbers on the board. Ali adds them and gets 15. Then he deletes one of the three numbers and replaces it with 3. Resi multiplies the three numbers and gets 36. Which numbers could Ali have deleted?
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Answer: B — either 7 or 8
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Hint 1 of 2
After swapping one number for 3, the product is 36, so the two untouched numbers multiply to 12.
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Hint 2 of 2
List the single-digit pairs that multiply to 12, then use the sum 15 to find the deleted number in each case.
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Approach: the two kept numbers multiply to 12; use the sum to recover the deleted one
- After replacing one number with 3, Resi's product 36 = 3 × (the two numbers Ali kept), so the kept pair multiplies to 12.
- Single-digit pairs with product 12 are 3 × 4 and 2 × 6.
- Since all three originals add to 15: if the kept pair is 3 and 4 (sum 7), the deleted number is 15 − 7 = 8; if the kept pair is 2 and 6 (sum 8), the deleted number is 15 − 8 = 7.
- So Ali could have deleted either 7 or 8.
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