Question

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?

Answer 1

Multiplying both sides of the congruence 2x≡y+5 (mod 9) by 5 gives 10x≡5y+25(mod 9),then reducing both sides modulo 9 gives x≡5y+7(mod 9).Thus, the product of the blanks is 5*7=35.