AMC 10 · 2025 · #4

Grade 8 number-theory
Archetype Small-Domain Constrained Search └ Sub-archetype Factor / Divisibility Constraint
base-conversiondigit-decompositionlinear-diophantine convert-to-algebra ↑ Prerequisites: base-conversion
📏 Short solution 💡 2 insights
Problem
A two-digit numeral a b read in base seven has the same value as the two-digit numeral b a read in base nine. Find the sum a+b of its two digits.

Pick an answer.

(A)
7
(B)
9
(C)
10
(D)
11
(E)
14
How to solve
Strategy Introduce a Variable

The digits a and b are the unknowns, so Tool #4 (Introduce a Variable) names them and turns the place-value meaning of each numeral into an equation. Solving that equation only pins down the ratio of the digits, so Tool #6 (Guess and Check) tests the small digit pairs that fit, and Tool #3 (Eliminate Possibilities) uses the base-seven digit limit (a,b ≤ 6) to throw out the larger pair that also fits the ratio but is an illegal digit.

1STEP 1

Write each numeral by place value

Expand both numerals by place value.

a b_ 7=7a+b, b a_ 9=9b+a
2STEP 2

Set the values equal and simplify

It simplifies to three a equals four b.

7a+b=9b+a → 6a=8b → 3a=4b
3STEP 3

Find the digits and add

Four and three are the only digits, so the sum is 7.

3a=4b→ a:b=4:3→ (a,b)=(4,3); a+b=7 → (A)
Answer
7
Both numerals evaluate to 31, so the setup is consistent: 7·4+3=31 and 9·3+4=31. The digits 4 and 3 are each below 7, so they are legal in base seven (and base nine), and the leading digit 4 is nonzero, so a b really is a two-digit number. The sum 4+3=7 matches (A). The larger ratio pair (8,6) would give 8+6=14, exactly the trap in (E), but 8 is not a base-seven digit, so it is correctly rejected.
💡Key takeaway

A two-digit number in any base is just the base times the first digit plus the second digit, so writing that out turns a base puzzle into a simple equation.

  • Write each numeral by place value
  • Set the values equal and simplify
  • Find the digits and add

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