Question

Show that m − 1 is a factor of m²¹ − 1 and m²² − 1.

Answer 1

Let p(m) = m²¹ − 1 and q(m) = m²² − 1.

Divisor = m − 1

Now, 

p(1) = (1)²¹ − 1

= 1 − 1

= 0

Therefore, by factor theorem (m − 1) is a factor of p(m) = m²¹ − 1.

Also, 

q(1) = (1)²² − 1

= 1 − 1

= 0

Therefore, by factor theorem (m − 1) is a factor of q(m) = m²² − 1.

Hence, (m − 1) is a factor of m²¹ − 1 and m²² − 1.