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(i)Y is a positive integer.
(ii) Y-1 is divisible by 5;
(iii) Y is divisible by 2,3,4,6,7,8 and 9.

What is the smallest Y?.

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 Another approach | Comment 2 of 5 |
The LCM of all integers 2-9 is 2520.  Then Y*(Y-1) = 2520*n, which implies Y = (1 +/- sqrt[1+10080*n])/2

The first n which makes Y an integer is n=20 yielding Y=225 and Y-1=224.  But that has Y as a multiple of 3,5,9 and Y-1 as a multiple of 2,4,7,8 and neither a multiple of 6.

So I keep going until I find n=101 (the fifth n that yields integer Y) which yields Y=505 and Y-1=504.  This almost works, Y is a multiple of 5 and Y-1 is a multiple of 2,3,4,6,7,8,9 - the exact opposite of what the problem asks.

Keep going until I finally find n=1612 (the eleventh n) which yields Y=2016 and Y-1=2015.  This is what we want, Y is a multiple of 2,3,4,6,7,8,9 and Y-1 is a multiple of 5.

This is obviously slower than Steve's direct approach.

 Posted by Brian Smith on 2017-04-04 12:25:42

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