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# What is the value of x if 823519 divided by x produces a remainder that is three times the remainder obtained by dividing 274658 by x and x is greater than one?

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## A number a power of a variable or a product of the two is a monomial while a polynomial is the of monomials

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Let k be the whole number we are searching for.

Let 823519 divided by k be n, remainder 3r

and 274658 divided by k be m, remainder r

Then

823519 = n k + 3r and

274658 = m k + r

Multiply the second equation by three and subtract the first:

455 = (3m - n) k

So k is a factor of 455.

That means it is 1,5,7,13,35,65,91 or 455

I can write 823519 as 1809*455 + 424

Each time I divide this by one of the eight possibilities for k the 1809*455 part will

go exactly and not change the remainder. so I only need to look at the 424 part.

Listing the possibilities:

424 divided by 1 has a remainder of 0

424 divided by 5 has a remainder of 4

424 divided by 7 has a remainder of 4

424 divided by 13 has a remainder of 8

424 divided by 35 has a remainder of 179

424 divided by 65 has a remainder of 34

424 divided by 91 has a remainder of 60

424 divided by 455 has a remainder of 424

The 1 and 91 choices are the only ones which leave a remainder which is a multiple of 3

274658 divided by 91 has a remainder of 20.  Earn +20 pts  