Problem 21 · 2017 Math Kangaroo
Stretch
Number Theory
place-value
How many positive whole numbers have the property that, if you delete the last digit, you obtain a new number that is exactly equal to 114 of the original number?
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Answer: C — 2
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Hint 1 of 2
Deleting the last digit of N leaves the number formed by the other digits; call that part a and the last digit d.
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Hint 2 of 2
Write N = 10a + d and set a = N/14, then see which digits d are possible.
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Approach: set up the place-value equation and solve for valid digits
- Let the number be 10a + d, where a is what remains after deleting the last digit d.
- The condition a = (10a + d)/14 gives 14a = 10a + d, so 4a = d.
- Since d is a single digit, a = 1 (d = 4, number 14) or a = 2 (d = 8, number 28).
- That is 2 such numbers.
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