Ian Stewart wrote:If a and b are both divisible by 30, then each is divisible by 2, 3 and 5. Since 30 is the largest common factor of a and b, then the fourth prime which is a factor of a cannot be a factor of b. Thus:

a = 2*3*5*p
b = 2*3*5*q*r

where p, q and r are distinct primes larger than 5. Thus,

ab = (2^2)*(3^2)*(5^2)*p*q*r

This is the prime factorization of ab. To count the number of divisors of ab, we add one to each power in the prime factorization and multiply:

3*3*3*2*2*2 = 216

Ian Stewart wrote:If a and b are both divisible by 30, then each is divisible by 2, 3 and 5. Since 30 is the largest common factor of a and b, then the fourth prime which is a factor of a cannot be a factor of b. Thus:

a = 2*3*5*p
b = 2*3*5*q*r

where p, q and r are distinct primes larger than 5. Thus,

ab = (2^2)*(3^2)*(5^2)*p*q*r

This is the prime factorization of ab. To count the number of divisors of ab, we add one to each power in the prime factorization and multiply:

3*3*3*2*2*2 = 216

pepeprepa wrote:You want to know the number of different factors of 3400
3400 = 2^3 * 5^2 * 17
X is the number of different factors of 3400
X=(3+1)*(2+1)*(1+1)=24

The number of different factors of 3400 is 24