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497= 71 x 7 rem 3
496 = 62 x 8 rem 4
495 = 55x 9 rem 5

so go back in 7s from 497 ....490,483 etc..
go back in 8s from 496...488, 480 etc
go back in 9s from 495 ....486, 477 etc until you arrive at the first number in all 3 lists

500?

I agree with Steve

icauser44 forget decimal places - the `remainder` is simply what`s left ... 71 x 7 = 497 which is 3 less than 500!

icauser44 you showed the correct working out in post 6!

Sorry, I meant post 5.

icauser44

I don`t know if they teach fractions nowadays ... but your 0.4285 is simply 3/7

My reasoning is:

If there were none left over in each case, the answer would be the Lowest Common Multiple of 7,8,9 = 504

Since there are some left over, the answer will be a bit more or a bit less than 504 . . the question tells us that it is 500 or less

The answer must be even number . .

By trial and error I started at 500, and was prepared to go 498,496 etc

But 500 was . . BINGO!

No need for trial and error - statement of problem implies number is 4 less than multiple of 7, 8 and 9, which is 504 - so answer is 500

Where does the 4 come from, please?