Problem 25 · 1997 AJHSME
Stretch
Number Theory
units-digitcyclicity
All the even numbers from 2 to 98 inclusive, except those ending in 0, are multiplied together. What is the units digit of the product?
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Answer: D — 6.
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Hint 1 of 2
A units digit only depends on the units digits of the factors — the tens never reach the ones place. So 12, 22, 32, … all behave like '2', and the long list shrinks to a repeating pattern of 2, 4, 6, 8.
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Hint 2 of 2
For a units digit of a huge product, replace each factor by its units digit, group the repeats, and use the fact that units digits of powers cycle.
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Approach: reduce to units digits, group, then use power cyclicity
- Excluding multiples of 10, each block of ten (2,4,6,8 then 12,14,16,18 …) contributes the units digits 2, 4, 6, 8. Their product ends like 2 × 4 × 6 × 8 = 384, i.e. units digit 4.
- From 2 to 98 there are 10 such blocks, so the overall units digit is that of 4¹⁰.
- Powers of 4 cycle 4, 6, 4, 6, …: 4² = 16 ends in 6, and any power of 6 ends in 6, so 4¹⁰ = (4²)⁵ ends in 6.
- You'll see it again: units digits of nⁿ-style products are tamed by (1) keeping only units digits and (2) exploiting their short repeating cycle — never multiply the giant number out.
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