Problem 22 · 2014 AMC 8
Medium
Algebra & Patterns
place-value-algebra
A 2-digit number is such that the product of the digits plus the sum of the digits is equal to the number. What is the units digit of the number?
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Answer: E — Units digit 9.
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Hint 1 of 2
The notice is that "digits" means place value: a 2-digit number is 10×(tens) + (units). Write the condition that way and watch how much cancels.
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Hint 2 of 2
The units digit b appears on both sides and vanishes; so does most of the rest, leaving a tiny equation that pins down one digit.
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Approach: place value, then cancel
- Write the number as 10a + b (tens digit a, units digit b). The condition "product + sum = number" becomes ab + a + b = 10a + b.
- The b on each side cancels, leaving ab = 9a.
- Since a ≠ 0 (it's the leading digit), divide by a: b = 9. The units digit is forced — the tens digit can be anything.
- Check: 1·9 + 1 + 9 = 19 ✓ (and 29, 39, … all work too).
- Why this transfers: turning "the digits" into 10a + b converts a word puzzle into algebra, and the magic here is that the answer doesn't depend on a at all — a sign the question only ever cared about the units digit.
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