- #1

Math100

- 756

- 204

- Homework Statement
- Assuming that ## 495 ## divides ## 273x49y5 ##, obtain the digits ## x ## and ## y ##.

- Relevant Equations
- None.

Observe that ## 495=5\cdot 9\cdot 11 ##.

This means ## 9\mid 273x49y5 ## and ## 11\mid 273x49y5 ##.

Then ## 9\mid (2+7+3+x+4+9+y+5)\implies 9\mid (x+y+30)\implies x+y=6, 15 ## and ##11\mid (2-7+3-x+4-9+y-5)\implies 11\mid (y-x-1)\implies y-x=1 ##.

Now we compute these two systems of equations shown below:

##\{x+y=6, y-x=1\}## and ##\{x+y=15, y-x=1\}##

Thus ## x=7 ## and ## y=8 ##.

Therefore, the digits ## x ## and ## y ## are ## 7 ## and ## 8 ##.

This means ## 9\mid 273x49y5 ## and ## 11\mid 273x49y5 ##.

Then ## 9\mid (2+7+3+x+4+9+y+5)\implies 9\mid (x+y+30)\implies x+y=6, 15 ## and ##11\mid (2-7+3-x+4-9+y-5)\implies 11\mid (y-x-1)\implies y-x=1 ##.

Now we compute these two systems of equations shown below:

##\{x+y=6, y-x=1\}## and ##\{x+y=15, y-x=1\}##

Thus ## x=7 ## and ## y=8 ##.

Therefore, the digits ## x ## and ## y ## are ## 7 ## and ## 8 ##.

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