Problem number 20. How many integers are factors of 2 to the power of 20 times 3 to the power of 23 but are not factors of 2 to the power of 10 times 3 to the power of 20? Note that 2 to the 10 times 3 to the 20 is a factor of 2 to the 20 times 3 to the 23. So, all factors of 2 to the 10, 3 to the 20 are also factors of 2 to the 20 times 3 to the 23. So, from here we know that we can find the number of factors of 2 to the power of 20 times 3 to the power of 23 and subtract the number of factors of 2 to the power of 10 times 3 to the power of 20. So, to find the number of factors of a number, for example, the number of factors of 2 to the 20 times 3 to the 23, you just add one to each of the exponents and then multiply them. So, that has 21 times 24 factors and 2 to the power of 10 times 3 to the power of 20 has 11 times 21 factors. So, now using our complementary factors or complementary counting skills, we can just do 21 times 24 minus 11 times 21 and that gives us 21 times 24 minus 11 and that gives us 273 as a correct answer and choice C is that.

factors

integers

exponents

complementary counting

mathematics