Difference between revisions of "1997 AJHSME Problems/Problem 25"
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Latest revision as of 23:28, 4 July 2013
Problem
All of the even numbers from 2 to 98 inclusive, excluding those ending in 0, are multiplied together. What is the rightmost digit (the units digit) of the product?
Solution
All the tens digits of the product will be irrelevant to finding the units digit. Thus, we are searching for the units digit of
There will be groups of numbers. The number now can be rewritten as
Simplifiying the inside, we get
Again, we can disregard the tens and hundreds digit of , since we only want the units digit of the number, leaving .
Now, we try to find a pattern to the units digit of . To compute this quickly, we once again discard all tens digits and higher.
.
, discard the .
, discard the .
Those equalities are, in reality, congruences .
Thus, the pattern of the units digits is . The cycle repeats so that term is the same as term . The tenth number in the cycle is , giving an answer of
See Also
1997 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
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