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April 17, 2015

Posted by **kala** on Friday, June 8, 2012 at 10:58am.

if counted in pairs, one will remain

if counted in threes, two will remain

if counted in fours, three will remain

if counted in fives, four will remain

if counted in sixes, five will remain

if counted in sevens, nothing will remain

what is the number?

- maths -
**MathMate**, Friday, June 8, 2012 at 12:12pmThis is a problem in number theory and related to the Chinese remainder theorem.

I do not know at what level you're working on. However, this problem is probably made to be solved without the use of advanced theory. Here's an approach that you can apply to other problems of the same nature.

We have the following conditions:

1. if counted in pairs, one will remain

2. if counted in threes, two will remain

3. if counted in fours, three will remain

4. if counted in fives, four will remain

5. if counted in sixes, five will remain

6. if counted in sevens, nothing will remain

Conditions 1 to 5 "happen" to be one less than a multiple of the counting number. Therefore, we can find a solution to conditions 1 to 5 by finding the LCM of the corresponding numbers 2,3,4,5,6*and subtract 1*.

Since LCM(2,3,4,5,6)=3*4*5=60, we see immediately that 59 satisfies all of conditions 1 to 5. You should check this for yourself.

We can also see that for any positive integer k, k*LCM-1 will also share the same property. For example, for k=3, 60*3-1=179 is also a solution to conditions 1-5. Also, check this for yourself.

We conclude therefore, to satisfy condition 6, we only need to find (by trial and error), a value of k such that 7 divides (60*k-1), in which case (60k-1) will be the required number.

I will leave it to you to complete the problem. Please post any time if you have questions.

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