Question

Whitney buys two submarine sandwiches for an after-school party. One sandwich is 45 inches long and the other sandwich is 20 inches long. Whitney wants to cut servings of the same length from both sandwiches, without having any sandwich left over. What is the greatest length serving that Whitney can cut?

Answer 1

5 inches

Explanation:

Whitney wants the greatest length that is an even divisor of both 45 inches and 20 inches.

Greatest common factor

The length Whitney is looking for is called the "greatest common divisor" (GCD) or "greatest common factor" (GCF) of the two numbers 45 and 20. There are a couple of different ways this can be found.

Factoring

The prime factors of the two numbers are ...

45 = 3×3×5

20 = 2×2×5

The only factor common to the two products is 5. It is the greatest common factor.

Whitney can cut lengths of 5 inches.

Euclid's algorithm

Another way to find the GCD is to use Euclid's algorithm. That has you ...

• Divide the larger number by the smaller and keep the remainder.
• If the remainder is not zero, replace the larger number with it and repeat.
• If the remainder is zero, the smaller number is the GCD.

Using this algorithm, we have ...

45/20 = 2 r 5

20/5 = 4 r 0 . . . . . The GCD is 5

Whitney can cut lengths of 5 inches.