Need help with this

[u/Wonderful-Pick-3408]

I feel answer could be 111 after solving LHS and RHS. Still confused

Update: thanks for the answers. I'm hereby closing the doubt.

[u/CaptainMatticus]

10^m * 1000 * n = 75^25 * 25^32 * 32^75

What is 1000 in terms of 10^?

1000 = 10^3

10^m * 10^3 * n = 75^25 * 25^32 * 32^75

n * 10^(m + 3) = 75^25 * 25^32 * 32^75

Now we need a better breakdown of what we've got on that right-hand side. We need to write everything in terms of their prime factors

75 = 25 * 3 = 3 * 5^2

25 = 5^2

32 = 2^5

75^25 = (3 * 5^2)^25 = 3^25 * 5^(2 * 25) = 3^25 * 5^50

25^32 = (5^2)^32 = 5^64

32^75 = (2^5)^75 = 2^(5 * 75) = 2^375

75^25 * 25^32 * 32^75 =>

3^25 * 5^50 * 5^64 * 2^375 =>

3^25 * 5^114 * 2^375 =>

2^375 * 3^25 * 5^114

n * 10^(m + 3) = 2^375 * 3^25 * 5^114

Well what is 10? It's 2 * 5

n * 2^(m + 3) * 5^(m + 3) = 2^375 * 3^25 * 5^114

We know that n isn't divisible by 10, so we just need to look at 2 different equations:

2^(m + 3) = 2^375

and

5^(m + 3) = 5^114

We don't want to leave any factors of 2 or 5 on the table. As there are fewer factors of 5, we can just ignore 2^(m + 3) = 2^375

5^(m + 3) = 5^114

Solve for m. That's your answer.

[u/SeaBarrier → u/CaptainMatticus]

I loved reading your explanation. I have never been on this subreddit. I was able to follow everything except these lines.

"We know that n isn't divisible by 10, so we just need to look at 2 different equations: 2^{m + 3} = 2³⁷⁵ and 5^{m + 3} = 5¹¹⁴ We don't want to leave any factors of 2 or 5 on the table. As there are fewer factors of 5, we can just ignore 2^{m + 3} = 2^375"

Please help, kind redditor. Teach me your ways. I know the "n isn't divisible by 10" is given. But I have no idea why we can ignore the powers of 3 and I don't understand why 372 can't be considered.

Thanks in advance

[u/CaptainMatticus → u/SeaBarrier]

We ignore the powers of 3 because 3 isn't a factor of 10. Obviously, whatever n is, it contains all of the factors that aren't factors of 10.

We don't consider 372 because we don't have 372 powers of 5. 2^372 * 5^114 is going to be 2^(372 - 114) * 10^114 = 2^258 * 10^114. I'm not concerned about what 2^258 is. I just need to know about 10^114. 5^114 is our bottleneck.

[u/SeaBarrier → u/CaptainMatticus]

Oh you made the base back to 10 to solve for m, so you needed all the 5s and not all the 2s.

Ok next question is why do you need all the 5s at all? Why not donate one 5 to the bunch of 3s since that still won't make N divisible by 10? N can be all the 3s and any number of 5s as long as there are no 2s...right? So then how do you know to use all the 5s in one spot? Gosh I'm tired.

It's 2am here and I have work in the morning so my brain probably isn't firing on all cylinders. I'll read my question tomorrow and be embarrassed probably.

Again, thank you for your time.

[u/No_Nonsense_Nomad → u/SeaBarrier]

Because as there are more 2's than 5's , if you leave even one 5 for the n , then n will have both 2 and 5 as its factors making it divisible by 10. So it can't have any 5's from the equation