Problem 9 · 2024 Math Kangaroo
Medium
Number Theory
factorization
Noah starts with the number 1 and multiplies it with either 6 or 10. He then multiplies the result again by either 6 or 10. He repeats this process several times. Which of the following numbers can he not obtain in this way?
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Answer: B — \(2^{90}\cdot 3^{30}\cdot 5^{80}\)
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Hint 1 of 2
Each multiplication by 6 or 10 always adds one factor of 2, plus either a 3 or a 5.
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Hint 2 of 2
So in the final number the power of 2 must equal the powers of 3 and 5 added together — check which option breaks this.
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Approach: count factors of 2 against 3 and 5
- 6 = 2·3 and 10 = 2·5, so each step adds exactly one 2 and either one 3 or one 5.
- After several steps the power of 2 equals (power of 3) + (power of 5).
- Option B has 2⁹⁰·3³⁰·5⁸⁰, but 30 + 80 = 110 ≠ 90.
- So B cannot be obtained.
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