Answer: The number is: 349 .______________________________
Explanation:
____________
Let "x" represent the number we wish to find:
____________________________
Given: "When a number ("x") is divided by "13", the remainder is "11".
________________________________
As such,
_________________
→ x = 13p + 11 ;
_______________________
Given: "When the same number ("x") is divided by "17", the remainder is "9".
As such:
______________________
→ x = 17q + 9
________________________
Since x = 13p + 11 ; and x = 17q + 9 ;
___________________________
→ 13p + 11 = 17q + 9 ; → Subtract "13p" and subtract "11" from EACH side of the equation;
__________________________
→ 13p + 11 − 13p − 11 = 17q − 13p + 9 − 11 ;
to get:
___________________________________
→ 0 = 17q − 13p − 2 ; ↔ 17q − 13p − 2 = 0 ;
→ Add "2" to each side of the equation;
→ 17q − 13p − 2 + 2 = 0 + 2 ;
→ to get: 17q − 13p = 2 ;
_____________________
Now, solve for "q", by isolating "q" on one side of the equation:
______________________
→ We have: 17q − 13p = 2 ;
→ Add "13p" to EACH side of the equation: 17q − 13p + 13p = 2 + 13p ;
→ to get: 17q = 2 + 13p ;
→Now, divide EACH side of the equation by "17"; to isolate "q" on one side of the question, and to solve for "q" ;
____________________________
→ 17q / 17 = (2 + 13p) / 17 ;
→ q = (2 + 13p) /17
The value of "p" for which "[(2 + 13p) /17]" i<span>s a whole number is:
"26" (p = 26) ; How do we get this? We know that "q" should be greater than "1", so we can start with "p = 2"; we get: 13*p = 13*2 = 26, which happens to be a whole number, "26", at the lowest possible value of "p". So we do not have to work with more complicated methods.
</span><span>
___________________
So, since we have:
___________________
</span>x = 13p + 11 ;
→ We can plug in our value of "26" for "p", and solve for "x" ;
→ x = (13*26) + 11 ;
→ x = (338) + 11 ;
→ x = 349; which is our answer.
_____________________________
Let us check our answer.
_____________________
Given:
<span>"When a number is divided by 13 the remainder is 11."
So, when 349 is divided by 13, is the remainder, "11"??
(i.e. 349 </span>÷ 13 =? a whole number value PLUS (11/13)???
→ 349 ÷ 13 = 26.8461538461538462 ;
and: 11/13 = 0.8461538461538462 ; so, YES!
_________________________________________
When 349 is divided by 17, is the remainder "9"?
(i.e. does 349 ÷ 17 = a whole number value, PLUS (9/17)??
___________________
→ 349 ÷ 17 = 20.5294117647058824 ;
→ 9 ÷ 17 = 0.5294117647058824; so, YES!
________________________________________
