2025 AMC 12B Problem 4

Attempt Problem 4 of the 2025 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2025 AMC 12B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

The value of the two-digit number $\underline{a}\,\underline{b}$ in base seven equals the value of the two-digit number $\underline{b}\,\underline{a}$ in base nine. What is $a + b?$

$7$

$9$

$10$

$11$

$14$

Answer: A

Concepts:number base linear equation

Difficulty rating: 1200

Solution:

Setting $7a + b = 9b + a$ gives $6a = 8b,$ so $3a = 4b.$ Thus $a=4t$ and $b=3t.$ Because both are base-seven digits and $a\ne0,$ only $t=1$ works. Hence $a=4, b=3,$ and indeed $43_7 = 31 = 34_9.$ Therefore $a+b=7.$

Thus, the correct answer is A.

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