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

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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.
  • #1
Math100
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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 ##.
 
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  • #2
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.
 
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  • #3
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.
 
  • #4
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.
 
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Related to Obtain the digits ## x ## and ## y ##

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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