Competition · AMC preparation · step 4 of 4
AMC 8 · 2005 · #18
Grade 4 number-theoryPick an answer.
AMC 8 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Counting three-digit multiples of 13 directly is awkward — the multiples (104, 117, 130, …) are not the numbers we want to count. Tool #9 (Easier Related Problem) replaces the task with something much simpler: each three-digit multiple has the form 13k, so we only need to count the values of k that make 13k a three-digit number. That turns the question into "how many integers k satisfy 100 ≤ 13k ≤ 999?" — a plain integer-counting problem. Tool #6 (Guess and Check) is the right way to find the two endpoints of k: try a few values near 100/13 and 999/13 until 13k first lands in, and last lands inside, the three-digit window. Once the endpoints are pinned down, the count is just (last - first) + 1.
Rewrite using multiples of 13
Every three-digit multiple of 13 is 13k, so count the integers k with 100 ≤ 13k ≤ 999 instead.
Every multiple of 13 is 13 × something. So instead of hunting for three-digit multiples, hunt for the multipliers k — they form a tidy block of consecutive integers.
4.OA.B.4Solve An Easier Related ProblemFind the smallest k
Test k = 8: 13 × 8 = 104 is the first three-digit multiple, so the smallest k is 8.
Dividing 100 by 13 tells you roughly where the boundary sits; the integer just above the quotient is your first valid k.
4.NBT.B.6Guess And CheckFind the largest k
13 × 76 = 988 fits but 13 × 77 = 1001 has four digits, so the largest k is 76.
Same idea at the top end: divide 999 by 13 and round down. Two cheap multiplications confirm you have not slipped past the boundary.
4.NBT.B.6Guess And CheckCount from 8 to 76
The valid k run 8, 9, …, 76, so the count is (76 - 8) + 1 = 69.
When you count fenceposts from 8 to 76, you get one more than the difference. The +1 catches the first post you would otherwise miss.
The number of integers k from 8 to 76 inclusive is (76 - 8) + 1.
▸ Why?
Two things fix the count: the subtraction 76 - 8 measures how many single steps of 1 separate the first value from the last, and the +1 adds back the starting value itself, which stepping from one end to the other never counts.
▸ Why?
Walking from 8 up to 76 one integer at a time takes 76 - 8 = 68 steps, and each step lands on exactly one new integer, so the steps match one-for-one with the 68 values that come after the start.
▸ Why?
Those 68 stepped-to values together with the single starting value 8 make up the whole list with no gap and no overlap, so the total count is the 68 plus that 1.
Instead of counting three-digit multiples of 13, count the multipliers k. The smallest is 8 (since 13 × 8 = 104), the largest is 76 (since 13 × 76 = 988), and the integers from 8 to 76 number (76 - 8) + 1 = 69.
- Rewrite using multiples of 13
- Find the smallest k
- Find the largest k
- Count from 8 to 76
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