Can someone help me out i am confused on how they solved this

https://math.stackexchange.com/questions/1780856/question-from-mathcounts-competition

this is hard if you solve it you are very smart. It is already is explained but i dont get it

SketchyPerson Nov 8, 2019

#1**+2 **

Hi Sketchy,

I have used the same logic as the other person but i have added a few extra words.

So if you can understand.

Feel free to ask questions or comment.

The least positive integer that is divisible by 2,3,4 and 5 and is also a perfect square, perfect cube, 4th power, and 5th power, can be written in the form a^b for positive integers a and b.

What is the least possible value of a+b?

The answer is 90

The number is a^b where a is the smallest number that will work.

For instance 4^3 is not ok because 4 = 2^2 so 4^3 can be written as 2^6

This is not worded well but more exact wording would probably just add to confusion anyway.

a^b is a square number so b is a multiple of 2

a^b is a cubic number so b is a multiple of 3

a^b is a power of 5 so b is a multiple of 5

a^b is a power of 4 so b is a multiple of 4 but it is already a multiple of 2 so we just need one more 2

That is

b=2*3*5*2 = 60 It could be bigger but that is the smallest it can be.

Now what number to the power of 60 is a multiple of 2,3,4 and 5

2^60 is **not** a multiple of 3 or 5 but it **is** a multiple of 2 and 4

(2*3)^60 is **not** a muliple of 5 but it **is** a muliple of 2,3 and 4

(2*3*5)^60 **is** a muliple of 2 and 3 and 4 and 5.

so the smallest values of a and b are 30 and 60 respectively

a+b = 30+60 =90

Melody Nov 9, 2019

#2

#5**0 **

Because I had started an answer but not finished it so i hid it until I had time to finish it later.

The ability to do this is one of the perks of being a moderator.

Ususally when there is less answers than is indicated it is becasue one or more of the answers has been hidden either automatically or by a moderator because it is inappropriate or because it needs to be checked for suitablility.

For instance:

Many posts with links are automatically blocked until a mod deems it suitable.

Melody
Nov 9, 2019

#7**+1 **

Hi Magic Kitten,

Mostly mods get rid of garbage, like really bad posts, and they tick off answers.

Just because they are ticked off soen't have to mean that the answer is right. We don't have time to check everything.

That is about it.

If you ever want to ask questions about a post always do so.

Even if it is ticked off you can still ask question. :)

After a period of time, questions get blocked so you can't answer any more but you only have to send a private message to a mod and they can unlock it for you.

Melody
Nov 9, 2019

#3**+1 **

To guarantee a number is a perfect square, cube, 4^{th} or 5^{th} power, we need to raise the number to the power of 3 × 4 × 5 = 60. We also want the number to be divisible by 2, 3, 4 and 5. If we raise 2 × 3 × 5 = 30 to any power, 2 or greater, then it will be divisible by these integers. So, 30^{60} would satisfy our conditions. The least possible value of a + b is therefore 30 + 60 = 90. ^{ }

Guest Nov 9, 2019