Advanced Placement (AP)

Question

When a number is divided by 13 the remainder is 11 . When the same number is divided by 17 remainder is 9. What is the number

2 Answer

  • the answer is 13*9+11=128 or 17*7+9=128
  • 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]" is 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.

    ___________________
    So, since we have: 
    ___________________
    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: 

    "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 
    ÷ 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!
    ________________________________________
NEWS TODAY