Oliver B.

asked • 04/30/17

Why can x only be a number in "35n + 17"

r => row of chairs
x => total amount of chairs
When you try to make 5 rows, there's 12 chairs left over.
When you try to make 7 rows, there's 4 chairs too little.
And that means:
"x - 12" must be evenly divisible by 5; 
"x + 4" must be evenly divisible by 7;
Find what x´s are possible
Numbers that are after subtracted by 12 evenly divisible by 5: 
17, 22, 27, 32, 37, 42, 47, 52, 57, 62, 67, 72, 77, 82, 87...
Numbers that are evenly divisible by 7 after 4 is added to that number:
3, 10, 17, 24, 31, 38, 45, 52, 59, 66, 73, 80, 87...
So as you can see, the numbers that fit into both the categories above are 17, 52 and 87. Each step is the size of 35 since if a number has to both be evenly divisible by 5 and 7 it has to have 5 and 7 as factors, or 35 as a factor. So that's why the size of each step is 35.
(As shown: 17+35 = 52; 52+35 = 87)
But why does it start at 17?
If you'd write an expression for every possible x it would be "17 + 35n".
This is what I don't understand. Where does this 17 come from? Why is 17 the first number that fits the requirements for x? It must have something to do with the - 12 and + 4, but I don't know what. I only solved the problem by testing, not with a methodical approach.
plz help me

Oliver B.

I don't really know why I used n. That's what they always use in school for formulas. Maybe n is more general? I dunno..
But why is 17 the smallest number to fit both criteria? For a number that has to be evenly divisible by 3 and 4 the smallest number that fit those criteria is 12. That's because 12 has both 3 and 4 as factors.
But when you when you change the criteria by including subtraction or addition the smallest number that fit both criteria changes. Is there a similar explanation for those cases? In this case a +4 and a -12 made the formula "35n+17" instead of just "35n". But what if it was a +5 instead of a +4? How does the formula change? If I were to try to solve that I'd have to try possible x's as I did here. But I want to be able to solve it intuitively. 


David W.

Oliver, what does the problem ask you to do? I can keep helping with your train of thought, but I'm not sure we're on the right track.


2 Answers By Expert Tutors


David W. answered • 04/30/17

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