K is the least positive integer divisible by all positive integers less than or equal to 10. Find the total number of factors of K?

Question

the answer is 46. I am not able to figure out how to arrive at that answer

Answer 1

Well, actually you can't arrive at the answer, because the answer should be 48.

The long way is to list all factors of 2520, viz. 1 through 10, 12,14,15, ... 2520

but the theory of numbers says that if you break them up into prime factors, viz $2^3 \cdot3^2 \cdot 5^1 \cdot 7^1$,

add 1 to each of the indices and multiply them out, you get the answer: $4\cdot3\cdot2\cdot2 = 48$

Look here to see all 48 factors

NOTE

The term proper factor should have been used in the question to get the answer of 46, but if you have been taught to exclude 1 & the number anyway, just subtract 2 from the number theory computation.

Answer 2

The least positive integer divisible by $1,2,3,4$ is $4\cdot 3 = 12$. Note that it is not $2\cdot 3\cdot 4 = 24$ - we don't need to include the $2$, because any integer that is divisible by $4$ is also divisible by $2$.

Now you need to generalize that idea to $1,2,\cdots,10$.