How do you find # of common factors between 2 numbers? e.g how many factors of 3 are between 100 and 200? Is there a quick way to calculate?

Question

How do you find # of common factors between 2 numbers? e.g how many factors of 3 are between 100 and 200? Is there a quick way to calculate?

Answer 1

Yes! There is absolutely a quick way to calculate this! (I assume you mean multiples, not factors, as the only factors of three are one and three, so there are none between 100 and 200.) All numbers that are multiples of three have digits that sum to multiples of three, so I know 102 is the smallest multiple of three in this range, since 1+0+2=3, and 3 is divisible by 3. 198 is the largest, since 1+9+8=18, and 18 is divisible by 3.

Knowing the first and last values, I used the arithmetic sum formula with a common difference, d=3 and first term=102:

a_n=3(n-1)+102.

Using 198 as a_n, the "current" or n^th term, I solved for n to find the number of terms:

198=3(n-1)+102. Subtract 102 from both sides to get

96=3(n-1). Divide by 3 to get

32=n-1. Add one to get

33=n

So 198 is the 33^rd term in this sequence of numbers, which are all the multiples of 3 between 100 and 200. That's how many multiples of three there are.

Also, if you're not familiar with arithmetic sequences, check out this intro in Purple Math: https://www.purplemath.com/modules/series3.htm

or, for something more comprehensive, search for "arithmetic sequences" on www.khanacademy.org.

Happy mathing!

Liz Z.