Problem 26 · 2017 Math Kangaroo
Stretch
Algebra & Patterns
arithmetic-sequencelast-digit
The number sequence 2, 3, 6, 8, 8, … is created by the following rule: the first two digits are 2 and 3. After that, every subsequent digit is the units digit of the product of the two previous digits. Which digit is the 2017th digit of the sequence?
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Answer: A — 2
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Hint 1 of 2
Each new term is the units digit of the product of the previous two, so the sequence soon repeats.
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Hint 2 of 2
Find the repeating block, then locate position 2017 within it.
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Approach: detect the cycle, then index into it
- The sequence is 2, 3, 6, 8, 8, 4, 2, 8, 6, 8, 8, 4, ... ; from the 6th term it cycles with period 6: (4, 2, 8, 6, 8, 8).
- Position 2017 is 2011 steps past the 6th term, and 2011 mod 6 = 1, picking the 2nd entry of the cycle.
- That entry is 2.
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