Question

Factor 40+64. Write your answer in the form a(b+c) where a is the GCF of 40 and 64.

Answer 1

To factor the expression \(40 + 64\), we first need to find the greatest common factor (GCF) of the numbers 40 and 64.

The prime factorization of 40 is:
\[
40 = 2^3 \times 5
\]

The prime factorization of 64 is:
\[
64 = 2^6
\]

To find the GCF, we take the lowest power of each common prime factor. The only prime factor common to both numbers is 2. The lowest power of 2 in the two factorizations is \(2^3\).

Thus, the GCF of 40 and 64 is:
\[
\text{GCF} = 2^3 = 8
\]

Next, we factor \(40 + 64\) using this GCF. We can express both numbers in terms of the GCF:

\[
40 = 8 \times 5 \quad \text{and} \quad 64 = 8 \times 8
\]

Now we rewrite the expression \(40 + 64\):
\[
40 + 64 = 8 \times 5 + 8 \times 8
\]
Factoring out the GCF (8):
\[
40 + 64 = 8(5 + 8)
\]

Now we simplify the expression inside the parentheses:
\[
5 + 8 = 13
\]

Thus, we have:
\[
40 + 64 = 8(13)
\]

Putting it all together, we write the final factored form:
\[
40 + 64 = 8(5 + 8) = 8(13)
\]

In summary, the answer in the form \(a(b + c)\) is:
\[
\boxed{8(5 + 8)}
\]