Even Factors with PF

There are two ways to do this.

Way 1

Find the total number of positive factors and subtract the number of odd positive factors.

# of positive factors $-$ # of odd factors $=$ # of even factors

• Example: How many even positive factors does $1$,$500$ have?
  • Prime Factorization: $=2^23^15^3$
  • # of positive factors: $3 \times 2 \times 4 = 24$
  • # of positive odd factors: $2 \times 4 = 8$
  • # of positive even factors: $24 -8=16$

Way 2

First prime factorize the number in question. Add $1$ to the exponent of every odd prime divisor, but do nothing to the exponent on the $2$. Multiply those numbers together.

• Example: How many even positive factors does $1$,$500$ have?
  • Prime Factorization: $=2^23^15^3$
  • add $1$ to exponent of each odd prime divisor: $(1+1), (3+1) = 2, 4$
  • Do nothing to the exponent on the $2$: it remains $2$
  • Multiply those numbers together: $2 \times 2 \times 4 = 16$