2011 AMC 10B Problem 5

P1-10~2 minPrint

Divisibility & Factors Bases & Digits Set up the equation/formula and compute; no special trick needed

In multiplying two positive integers $a$ and $b$ , Ron reversed the digits of the two-digit number $a$ . His erroneous product was $161$ . What is the correct value of the product of $a$ and $b$ ?

Show solution without answering

Key insight

Break 161 into factors: it is 7 times 23. The reversed number has two digits, so it was 23. That makes the real number 32 and the other number 7.

Solution

Ron's wrong product is (reversed $a$ ) times $b$ , which equals $161$ . Knowing the factors of $161$ will tell us what the reversed $a$ was.

Factor $161$ . It is odd, and its digits add to $8$ , so $2$ and $3$ do not divide it. It does not end in $0$ or $5$ . Try $7$ : $161 = 7 \times 23$ .
So $161$ can be split into two whole numbers only as $1 \times 161$ or $7 \times 23$ .
The reversed $a$ has two digits. The only two-digit factor in the list is $23$ . So the reversed $a$ is $23$ and $b = 7$ .
Flip $23$ back to get the real $a = 32$ .
The correct product is $32 \times 7 = 224$ .

The answer is $\boxed{\textbf{(E)}\ 224}$ .

Why this works

When a puzzle gives you the product of unknown whole numbers, factor it first. Here $161$ has so few factors that the "two-digit" clue leaves only one possibility.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Ron's product is (reversed $a$ ) $\times\ b = 161$ . Factor $161$ : it is odd, not divisible by $3$ or $5$ , and $161 = 7 \cdot 23$ . Its factorizations into two positive integers are $1 \cdot 161$ and $7 \cdot 23$ .

The reversed $a$ is a two-digit number, so it must be $23$ (the only two-digit factor), and $b = 7$ . Reversing $23$ back gives $a = 32$ .

The correct product is $32 \cdot 7 = 224$ , so the answer is $\boxed{\textbf{(E)}\ 224}$ .

Review in the AoPS books

Introduction to Number Theory, Chapter 1: Integers: The Basics (§1.4, §1.5)
Introduction to Number Theory, Chapter 8: Base Numbers
Introduction to Number Theory, Chapter 5: Divisor Problems (§5.2, §5.3)
Introduction to Number Theory, Chapter 9: Base Number Arithmetic