Competition · AMC preparation · step 4 of 4
AMC 8 · 2000 · #9
Grade 6 arithmeticnumber-theory
Pick an answer.
AMC 8 2000 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
There are only a few three-digit powers of each base, so Tool #2 (Make a Systematic List) gives us the complete set of candidates in seconds — multiply by 5 (or 2) until you leave the 100–999 range. Once both lists exist, Tool #6 (Guess and Check) handles the crossing constraint: the middle digit of the chosen 5-power has to match the first digit of the chosen 2-power. Listing both sides is faster and safer than any algebraic detour.
List three-digit powers of 5
Multiply 5 by 5 until you pass 999: the three-digit powers of 5 are 125 and 625.
The Grade 6 "whole-number exponents" skill: 5^k means multiply 5 by itself k times. Stop the list once you cross 1000.
6.EE.A.1Make A Systematic ListList three-digit powers of 2
The same way, the three-digit powers of 2 are 128, 256, and 512.
Same idea, base 2. There are exactly three candidates.
6.EE.A.1Make A Systematic ListRead the grid constraint
The grid ties 2 ACROSS's first digit to 1 DOWN's middle digit — and both 125 and 625 have middle digit 2.
Reading the tens place uses Grade 4 place value — and both possible 1 DOWN values agree, so the constraint is forced.
The hundreds digit of 2 ACROSS is forced to be 2.
▸ Why?
The one square the two entries share is at once the middle digit of 1 DOWN and the hundreds digit of 2 ACROSS, so those two digits are the same digit, and reading that square as the middle of 1 DOWN or as the front of 2 ACROSS must give the same value.
▸ Why?
Whichever allowed value 1 DOWN takes, its middle digit is 2.
▸ Why?
1 DOWN is a power of 5 — 5 multiplied by itself again and again — and each such product is strictly larger than the last (5, 25, 125, 625, 3125), so only 125 and 625 land in the three-digit range.
▸ Why?
In both 125 and 625 the digit standing in the tens place is 2, and picking out the tens place of a written number is exactly what place value lets us do.
Match the shared digit
Among 128, 256, 512, only 256 starts with 2, so 2 ACROSS = 256 and the outlined square is its units digit.
Match the first-digit clue against the small list — only one power of 2 survives.
4.NBT.A.2Guess And CheckWhen a puzzle hides a number behind a rule like "three-digit power of 5," list every option — there are usually only a few. Then the grid's shared cell does the rest of the work for you.
- List three-digit powers of 5
- List three-digit powers of 2
- Read the grid constraint
- Match the shared digit
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