SOLUTION: Determine the largest integer less than or equal to 2016 that leaves a remainder of 3 when divided by 7 and leaves a remainder of 4 when divided by 11.

Question 1055383: Determine the largest integer less than or equal to 2016 that leaves a remainder of 3 when divided by 7 and leaves a remainder of 4 when divided by 11.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
it looks like the number will be 1984.

i don't know if there's an easy way to do this.

this is how i did it.

divide 2016 by 7 and you get 288 with a remainder of 0.
subtract 4 from 2016 and you get 2012.
divide 2012 by 7 and you get 287 with a remainder of 3.
that's the largest number less than or equal to 2016 that is divisible by 7 with a remainder of 3.

divide 2016 by 11 and you get 183 with a remainder of 3.
add 1 to 2016 and you get 2017.
divide 2017 by 11 and you get 183 with a remainder of 4.
you got the remainder of 4 but 2017 is greater than 2016, so that's no good.
subtract 11 from 2017 and you get 2006.
divide 20006 by 11 and you get 182 with a remainder of 4.
that's the largest number less than or equal to 2016 that is divisible by 11 with a remainder of 4.

your two largest numbers are:

2012 and 2006.

you want to find the largest number that is divisible by both and leaves a remainder of 3 when divided by 7 and leaves a remainder of 4 when divided by 11.

work your way down each set.

2012 is reduced by 7 each time.
2006 is reduced by 11 each time.

you get:

2012 - 7 = 2005 - 7 = 1998 - 7 = 1991 - 7 = 1984

2006 - 11 = 1995 - 11 = 1984.

1984 is the magic number as best i can determine.

when it is divided by 7, you get 283 plus a remainder of 3.
when it is divided by 11, you get 180 plus a remainder of 4.

there may be an easier way to figure this out but i don't know it.

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