Whole Number to Exponents

Question

I need to find the exponents from $3240$. The answer sheet says $2^3\cdot3^4\cdot5$. But how do I get the exponents from my whole number?

Answer 1

For the powers of $2$; keep dividing by $2$ until you can no longer: 3240-> 1620 -> 810 -> 405 .. that's it: 405 is not divisible by $2$. So, you were able to divide by $2$ three times, hence the exponent of $2$ is $3$.

Now continue dividing by $3$ ...

Answer 2

Straight away we can identify that $3240$ is divisible by $10$, or equivalently $2\cdot 5$; so we can start with $3240=2\cdot 5\cdot 324$. Now, by using this divisibility test we can see that $324$ is divisible by $3$ precisely $108$ times. A further inspection sees that $108$ divides by $3$ again. So so far, we have $3240=2\cdot 5\cdot 3\cdot 3\cdot 36$. From here, we can swiftly conclude this by realising that $36=6^2=(2\cdot 3)^2=2^2\cdot 3^2$. Thus, putting everything together gives us $$3240=2\cdot 5\cdot 3\cdot 3\cdot 2^2\cdot 3^2=2^3\cdot3^4\cdot 5,$$ as desired.