Question

How do you write the sum of the number 48+14 as the product of their GCF and another sum?

Answer 1

Hint: We are given two numbers and we have to express their sum as a product of their GCF. For solving this question, we must know the definition of GCF. The full form of GCF is Greatest Common Factor, as the name suggests, the GCF of two numbers is the highest number that can divide both the numbers whose GCF we found. We will find the GCF of 48 and 14 by doing their prime factorization. After finding out the GCF we will express their sum as a product of the GCF and some other integers.

Complete step-by-step solution:
On the prime factorization of 48 and 14, we get –
$
  48 = 2 \times 2 \times 2 \times 2 \times 3 \\
  14 = 2 \times 7 \\
 $
We see that only 2 is the common factor of 48 and 14, so the GCF of 48 and 14 is 2.
Now, 48 can be written as $48 = 2 \times 24$ and 14 can be written as $14 = 2 \times 7$
On adding 48 and 14, we get –
$48 + 14 = 2 \times 24 + 2 \times 7$
Taking 2 as common, we get –
$
  48 + 14 = 2(24 + 7) \\
   \Rightarrow 48 + 14 = 2 \times 31 \\
 $
Hence the sum of the number 48+14 as the product of their GCF and another sum is written as $2 \times 31$ .

Note: GCF is also known as HCF (Highest Common Factor) and is equal to the product of the prime numbers that are common in the expansion of both the numbers. While taking 2 as common in the expression $48 + 14 = 2 \times 24 + 2 \times 7$ , we have used the distributive. According to the distributive property, $a(b + c) = ab + ac$ and vice versa.