2010 AMC 10B Problem 8

P1-10~2 minPrint

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

A ticket to a school play cost $x$ dollars, where $x$ is a whole number. A group of 9th graders buys tickets costing a total of $\$48$ , and a group of 10th graders buys tickets costing a total of $\$64$ . How many values for $x$ are possible?

Show solution without answering

Key insight

The ticket price must divide both 48 and 64 evenly. The prices that do are 1, 2, 4, 8 and 16, the numbers that divide 16.

Solution

Each group bought a whole number of tickets at $x$ dollars each, so $x$ must divide $48$ evenly and must also divide $64$ evenly.

1. List the numbers that divide $48$ :
$1,\ 2,\ 3,\ 4,\ 6,\ 8,\ 12,\ 16,\ 24,\ 48.$
2. List the numbers that divide $64$ :
$1,\ 2,\ 4,\ 8,\ 16,\ 32,\ 64.$
3. The numbers on both lists are $1, 2, 4, 8, 16$ . That is $5$ values.
4. Each one really works. For example, at $x = 16$ the 9th graders buy $3$ tickets and the 10th graders buy $4$ ; at $x = 1$ they buy $48$ and $64$ .

The answer is $\boxed{\textbf{(E)}\ 5}$ .

Why this works

The numbers that divide both $48$ and $64$ are exactly the numbers that divide their greatest common factor, $16$ . So "divides both" turns into "divides $16$ ", and $16 = 2^4$ has $5$ divisors: $1, 2, 4, 8, 16$ . Do not drop the smallest one ( $1$ ) or the largest one ( $16$ ).

Concise solution

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

Each group spent a whole number of tickets times $x$ dollars, so $x$ divides $48$ and $x$ divides $64$ . A common divisor of two numbers is a divisor of their greatest common divisor:
$\gcd(48, 64) = \gcd(2^4\cdot 3,\ 2^6) = 2^4 = 16.$
The divisors of $16$ are $1, 2, 4, 8, 16$ , five in all, and each one is a legitimate price (for example $x = 16$ means $3$ and $4$ tickets).

The answer is $\boxed{\textbf{(E)}\ 5}$ .

Review in the AoPS books

Introduction to Number Theory, Chapter 1: Integers: The Basics (§1.4, §1.5)
Introduction to Number Theory, Chapter 3: Multiples and Divisors (§3.3, §3.4, §3.7)
Introduction to Number Theory, Chapter 5: Divisor Problems (§5.2, §5.3)
Introduction to Number Theory, Chapter 4: Prime Factorization (§4.7)