Question

The number 4,000,000 has 63 positive integral factors. How do you find a and b, where 2^a 5^b is the product of all positive factors of 4,000,000?

Answer 1

`a=8` and `b=6`

Explanation:

`4000000=4xx10xx10xx10xx10xx10xx10`

= `2^2xx10^6`

= `2^2xx(2xx5)^6`

= `2^2xx2^6xx5^6`

= `2^(2+6)xx5^6`

= `2^8xx5^6`

Comparing it with `2^a5^b`, we get

`a=8` and `b=6`

Answer 2

`a=252` and `b=189`

Explanation:

`4000000 = 4 * 10^6 = 2^8*5^6`

So each of the positive integral factors of `4000000` is of the form `2^m*5^n` where `m in { 0, 1, 2, 3, 4, 5, 6, 7, 8 }` and `n in {0, 1, 2, 3, 4, 5, 6}`.

Note that:

`sum_(m=0)^8 m = 1/2 8 (8+1) = 36`

`sum_(n=0)^6 n = 1/2 6 (6+1) = 21`

We use these below...

The product of all the factors is:

`prod_(m=0)^8 (prod_(n=0)^6 2^m*5^n)`

`= prod_(m=0)^8 (2^(7m) prod_(n=0)^6 5^n)`

`= (prod_(m=0)^8 2^(7m)) (prod_(n=0)^6 5^n)^9`

`= 2^(7sum_(m=0)^8 m)*5^(9 sum_(n=0)^6 n`

`= 2^(7*36)*5^(9*21)`

`= 2^252 * 5^189`

So `a=252` and `b=189`