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Number Theory
Level
2
Hummus gives clues about his six-digit secret number:
- Clue 1: It is the same number if you read it from right to left.
- Clue 2: The number is a multiple of 9.
- Clue 3: Cross off the first and last digits. The only prime factor of the remaining four-digit number is 11.
What is Hummus’s six-digit number?
The answer is 513315.
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1 solution
Relevant wiki: Prime Factorization
Let's consider the clues one at a time, starting with the most restrictive.
Clue 3: The only prime factor of the 4-digit number is 1 1 , so the number is 1 1 × 1 1 or 1 1 × 1 1 × 1 1 or 1 1 × 1 1 × 1 1 × 1 1 , etc. Of these, only 1 1 × 1 1 × 1 1 = 1 3 3 1 has 4 digits, so the middle 4 digits are 1 3 3 1 .
Clue 1: The number reads the same right to left, so the first and last digits are the same.
Call the number A 1 3 3 1 A .
Clue 2: The number is a multiple of 9 , so the sum of its digits is a multiple of 9 . So, A + 1 + 3 + 3 + 1 + A = A + 8 + A = 2 A + 8 must equal 9 or 1 8 . No whole number A satisfies 2 A + 8 = 9 , but if 2 A + 8 = 1 8 ,then A = 5 .
Therefore, Hummus’s number is 5 1 3 3 1 5 .