Hi friends! Recently, I've been working on many mock MATHCOUNTS sessions and found myself stuck on this question:

Find the least positive integer b such that the value of the expression (4b + 9)(6b + 13) is a perfect square.

I'm just curious how you could solve these types of questions, so any help will be greatly appreciated!

Have a great one!

***

P.S. *"You’re stronger than any test, paper due, breakup, money issue, or any battle you’re facing. Keep your head up." *Anonymous

DreamingFox1214 Jan 24, 2022

#1**+5 **

First things first, we set the expression to be x^2.'

Next we can say 4b + 9 = x, and 6b + 13 = x.

Then using the transitive property of equality, 4b + 9 = 6b + 13.

b = -2

But the question is asking for a positive integer for b, and that does not have a valid way to quickly solve it. I checked this problem on MATHCOUNTS and it asks for a whole number/integer. so the answer is **b = -2**

proyaop Jan 24, 2022

#2**+3 **

If you're looking for the smallest positive b, then b = 18 gets you 99 squared.

Tiggsy Jan 24, 2022

#4**+2 **

Hey proyaop.....

You misread Tiggsy's answer .... '99 squared' not 99 so 18 works the result is 9801 =99^{2}

ElectricPavlov
Jan 24, 2022

#5**-1 **

**Attn: proyaop**

You have obviously changed this response so that it says the oposite of before.

Please do not do this.

You are actually responding to EP's comment when you made the edit but you did not acknowledge that fact.

The edit is fair but please acknowledge what you have edited and in this case your should have acknowledged EP's input.

------------

This is not a huge deal, you are making a lot of nice contributions here and we (other answerers) are very pleased to have you.

It is just something i wanted you to be aware of.

Melody
Jan 24, 2022

#6**+2 **

Ah! Thanks, proyaop! (and Tiggsy) That makes sense. Thanks for all your help!

DreamingFox1214 Jan 24, 2022