+239 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?
+983 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
