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

Answer:

35

Explanation:

2x == y+5 (mod 9)

2x * 5 = 10 * x = 1 * x = (y+5) * 5 (mod 9)

(5 is the inverse of 2 (mod 9))

x = 5y + 25 (mod 9)

x= 5y + 7 (mod 9)

5 * 7 = 35