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To find the number of positive factors for a given integer using Prime Factorization, follow these steps:
- first prime factorize the integer (eg: $3000 = 2^3 \times 3^1 \times 5^3$)
- add 1 to each exponent in the prime factorization (in this case, we would get $4$, $2$, $4$)
- multiply those numbers together (in this case $4 \times 2 \times 4 = 32$)
Here's another example. How many positive factors does $36$,$000$ have?
- Prime factorize: $2^53^25^3$
- Add $1$ to each exponent: $(5+1), (2+1), (3+1)=6,3,4$
- Multiply those numbes together:
- $6 \times 3 \times 4=72$
If a certain integer has only one prime factor, like $7^5$, simply add $1$ to the exponent $5+1=6$ to get the total number of positive factors.