# Obtain the digits ## x ## and ## y ##

• Math100
In summary, given that 495 divides 273x49y5, we can obtain the digits x=7 and y=8 by solving the equations x+y=6 and y-x=1.
Math100
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 ##.

Last edited by a moderator:
Delta2
Math100 said:
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:
## \left \{ \begin{align*} x+y=6 \\ y-x=1 \end{align*} \right \} ## and ##\left \{ \begin{align*} x+y=15 \\ y-x=1 \end{align*} \right \} ##
Thus ## x=7 ## and ## y=8 ##.
Therefore, the digits ## x ## and ## y ## are ## 7 ## and ## 8 ##.
Fixed some LaTeX in the quoted text above.

Last edited:
Delta2
SammyS said:
Fixed some LaTeX in the quoted text above.
Needed some additional fixing in LaTeX, or my browser for some reasons bugs seriously on displaying this message.

Math100 said:
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 ##
... since ##0\leq x,y \leq 9.##
Math100 said:
.
Now we compute these two systems of equations shown below:
##\{x+y=6, y-x=1\}## and ##\{x+y=15, y-x=1\}##
The first system yields ##2y=7## which has no integer solution, and the second ...
Math100 said:
Thus ## x=7 ## and ## y=8 ##.
Therefore, the digits ## x ## and ## y ## are ## 7 ## and ## 8 ##.

Correct.

Math100 and Delta2

## 1. How do I obtain the digits x and y?

The digits x and y can be obtained by breaking down the given number into its individual digits. For example, if the number is 123, x would be 1 and y would be 3.

## 2. What is the significance of obtaining the digits x and y?

Obtaining the digits x and y can help in various mathematical calculations and problem-solving. It can also provide insights into patterns and relationships within numbers.

## 3. Can the digits x and y be obtained for any number?

Yes, the digits x and y can be obtained for any number, regardless of its size or complexity. The process of obtaining the digits remains the same for all numbers.

## 4. Are there any techniques or shortcuts for obtaining the digits x and y?

There are various techniques and shortcuts, such as using divisibility rules, that can help in obtaining the digits x and y more quickly and efficiently. These techniques can be learned and practiced to improve speed and accuracy.

## 5. How can obtaining the digits x and y be useful in real life?

Obtaining the digits x and y can be useful in various real-life situations, such as calculating taxes, understanding data patterns, and solving mathematical problems. It can also be helpful in everyday tasks like checking bank account balances or identifying phone numbers.

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