A woodworker has two wooden boards. One board is 42 inches long, and the other is 70 inche… — ACT

A woodworker has two wooden boards. One board is $42$ inches long, and the other is $70$ inches long. She wants to cut both boards into shorter pieces of equal length, with no wood left over. What is the greatest possible length, in inches, of each shorter piece?

A
2
B
7
14Answer
D
28
E
210

Answer

To find the greatest possible length of the pieces, we need to find the greatest common factor (GCF) of the two board lengths,

42

and

70

. The prime factorization of

42

2 \times 3 \times 7

, and the prime factorization of

70

2 \times 5 \times 7

. The common prime factors are

2

and

7

. Multiplying these common prime factors gives the GCF:

2 \times 7 = 14

. Therefore, the greatest possible length of each piece is

14

inches.

Step-by-Step Solution

Identify the mathematical concept needed to solve the problem.

Since we need to cut two boards of lengths

42

and

70

into equal shorter pieces with no left over wood, we need to find a common factor of

42

and

70

. To find the greatest possible length, we must calculate their Greatest Common Factor (GCF).

This translates the word problem into a clear mathematical operation.

Find the prime factorizations of both numbers.

42 = 2 \times 3 \times 7

and

70 = 2 \times 5 \times 7

Prime factorization helps systematically identify all factors of the two numbers.

Determine the common prime factors and multiply them.

The common prime factors are

2

and

7

. Multiplying them gives

2 \times 7 = 14

The GCF is the product of all shared prime factors.

Key Concept

Greatest Common Factor (GCF)

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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