Dear Sir/Madam,
Good Day! I'm Benedict, a 16-years old student from the Philippines. I'm currently at 4th year high school We have an investigative project in mathematics. I have a problem with the presentation of my solution to the problem stated below:

By how much is 100 multiplied or divided, so that whether the number obtained is divided by 5 or 7, there will be a remainder of 1 and so that at the same time, the first quotient is greater by 4 than the second?

Please help me explain why the equation below is used:

        100x-1   100x-1
        ------ = ------ + 4
          5         7
		 

Why is 1 subtracted from 100x? What is the principle behind this? I have the difficulty in explaining to my teacher why the solution goes this way. Next week, i will have my defense and presentation for this problem and i am to present this to the whole schoo. Right now, this is my problem, on how to present this equation in an organized way.

Thank you very much!
Truly Yours,
Benedict

Hi Benedict,

You are to multiply 100 by some number so let that number be x. (It might turn out thet x is 1/y for some number y so you are actually dividing.) The instructions say that when you divide 100x by 5 the remainder is 1. That means that 1 less than 100x, when divided by 5, gives a remainder of zero. Thus

 ^{(100x - 1)}/₅ is an integer

Likewise

^{(100x - 1)}/₇ is an integer

The last part of the instructions say that the first of these integers is 4 more than the second.

Penny

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