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In the statement below, the two blanks can be filled by positive single-digit numbers in such a way that the statement is always true:
\($\text{If }2x\equiv y+5\ (\bmod\ 9)\text{, then }x\equiv \underline{\ \ \ }\,y+\underline{\ \ \ }\ (\bmod\ 9).\)

 

What is the product of the two digits that go in the blanks?

Rollingblade  May 17, 2018
 #1
avatar+945 
+2

Hey RB!

 

I'm not that good at number theory but here is what I have:

 

Multiplying both sides of the congruence:


\(2x\equiv y+5\pmod 9 \)

 

by 5 gives:

 

\(10x \equiv 5y+25\pmod 9,\)

 

then reducing both sides modulo 9 gives


\( x\equiv 5y+7\pmod 9. \)

 

Thus, the product of the blanks is \(5\cdot 7=\boxed{35}.\)

 

I hope this helped,

 

Gavin

GYanggg  May 17, 2018

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