Advanced Placement (AP), 28.01.2020 04:31 SpiritedAway7087

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

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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!

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Answer from: Quest

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