Problem 6 · AMC 8 Stretch
Stretch
Number Theory
adopt-a-different-point-of-viewpattern-recognition
Every entry below is a way to write the SAME number, but each in a different counting base, and the bases go DOWN by one each step. Find the rule and fill in the missing entry: \[10,\ 11,\ 12,\ 13,\ 14,\ \underline{\ \ },\ 100.\]
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Answer: 21 (the number 9 written in base 4)
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Hint 1 of 4
Don't read these as ordinary base-ten numbers. Try the idea that every entry stands for the SAME number, just written in a different base.
Still stuck? Show hint 2 →
Hint 2 of 4
Remember what a numeral means: '12' in base \(b\) means \(1 \times b + 2\). And '100' in base \(b\) means \(b \times b\).
Still stuck? Show hint 3 →
Hint 3 of 4
The last entry is \(100\). If \(100\) in some base equals our secret number, then the number is a perfect square. Try: the secret number is \(9\), and \(100\) is written in base \(3\) (since \(3 \times 3 = 9\)).
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Approach: Adopt a different point of view — every entry is one fixed number in shrinking bases
- Change your point of view: every entry is the SAME number, \(9\), just written in a different base, and the bases count DOWN by one each step (base \(9\), then \(8\), then \(7\), and so on).
- Check, going down in base: \(10\) in base \(9\) is \(1 \times 9 + 0 = 9\); \(11\) in base \(8\) is \(8 + 1 = 9\); \(12\) in base \(7\) is \(7 + 2 = 9\); \(13\) in base \(6\) is \(6 + 3 = 9\); \(14\) in base \(5\) is \(5 + 4 = 9\); and \(100\) in base \(3\) is \(3 \times 3 = 9\).
- The blank sits where the base is \(4\), so write \(9\) in base \(4\): \(9 = 2 \times 4 + 1\), which is the numeral \(21\).
- So the missing entry is \(21\) — the number \(9\) written in base \(4\).
Mark:
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