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When a natural number x is divided by 5, the remainder is 2. When a natural number y is divided by 5 the remainder is 4.The remainder is z when x+y is divided by 5. Find the value of (2z - 5)/3.

 Jul 6, 2020
 #1
avatar+25541 
+3

When a natural number x is divided by 5, the remainder is 2.
When a natural number y is divided by 5, the remainder is 4.
The remainder is z when x+y is divided by 5.
Find the value of \(\dfrac{2z - 5}{3}\).

 

\(\begin{array}{|lrcll|} \hline & x & \equiv & 2\pmod{5} \qquad (1) \\ & y & \equiv & 4\pmod{5} \qquad (2) \\ \hline (1)+(2): & x+y & \equiv & 2+4\pmod{5} \\ & x+y & \equiv & 6 \pmod{5} \\ & x+y & \equiv & 6-5 \pmod{5} \\ & x+y & \equiv & {\color{red}1} \pmod{5} \\ \hline & z &=& 1 \\\\ & && \mathbf{\dfrac{2z - 5}{3}} \\\\ & &=& \dfrac{2*1 - 5}{3} \\\\ & &=& \dfrac{-3}{3} \\\\ & &=& \mathbf{-1} \\ \hline \end{array}\)

 

laugh

 Jul 6, 2020
 #2
avatar+30920 
+1

The question implies the result is true for all such x and y, so we could choose x = 2, which has remainder 2 when divided by 5, and y = 4, which has remainder 4 when divided by 5.  Hence x + y = 6, which has remainder 1 when divided by 5, so z = 1.  Therefore (2z - 5)/3 = -1.

 Jul 6, 2020

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