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

 Jan 24, 2022

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





 Jan 24, 2022

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

 Jan 24, 2022

That also works Tiggsy smiley

proyaop  Jan 24, 2022
edited by proyaop  Jan 24, 2022

Hey proyaop.....

    You misread Tiggsy's answer ....   '99 squared'       not 99      so    18 works   the result is 9801 =992

ElectricPavlov  Jan 24, 2022

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
edited by Melody  Jan 25, 2022

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

 Jan 24, 2022
edited by DreamingFox1214  Jan 25, 2022

4 Online Users