Problem 11 · 2018 Math Kangaroo
Medium
Number Theory
last-digitplace-value
Martina multiplies two two-digit numbers and then paints over some of the digits, leaving \(?\,3 \times 2\,? = 3\,?\,2\). How big is the sum of the three digits that Martina has painted over?
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Answer: B — 6
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Hint 1 of 2
The units digit of the product comes only from the units digits of the two factors.
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Hint 2 of 2
Find the hidden units digit first, then the product is forced, revealing the other hidden digits.
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Approach: pin down digits using the units digit and size
- The numbers are (10a+3) and (20+b), product 3?2. For the product to end in 2, 3b must end in 2, so b = 4 and the second number is 24.
- Then (10a+3) · 24 lies between 300 and 392; a = 1 gives 13 · 24 = 312, which fits 3?2.
- The painted digits are 1, 4 and 1, whose sum is 6.
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