Understanding Why the Lowest Exponent Gives the Greatest Common Factor

Last Updated on June 7, 2026 by Rajeev Bagra

While solving recurrence relations such as

we often substitute

to obtain

The next step is usually to divide by

But why do we choose this particular power?

The Greatest Common Factor Principle

For powers of the same base, the greatest common factor (GCF) is obtained by taking the smallest exponent.

For example,

have GCF

Similarly,

have GCF

because is the smallest exponent.

A Quick Test for the GCF

A useful way to verify a GCF is to divide every term by it.

For example,

Notice two important things:

1. At least one term becomes .
2. No fractions or negative exponents appear.

This confirms that is the GCF.

What Happens If We Choose a Larger Factor?

Suppose we divide by

instead.

Then

A fraction appears, showing that is not a factor of every term and therefore cannot be the GCF.

Applying This to the Recurrence

Starting from

divide by the GCF,

to obtain

which leads directly to the characteristic equation

without introducing fractions.

Key Takeaway

When dealing with powers of the same base, the GCF is found by taking the smallest exponent. After dividing by the GCF, at least one term becomes and no term should contain negative exponents or fractions.