Final answer:
The modular equation 7 ≡ 5 mod 10 is solved by finding the smallest positive integer x such that 7 + x leaves a remainder of 5 when divided by 10. The solution is x = 8, since 15 (7 + 8) divided by 10 leaves a remainder of 5.
Step-by-step explanation:
To solve the modular equation 7 ≡ 5 mod 10, we need to find the smallest positive integer x such that when 7 is divided by 10, the remainder is 5. The solution to this particular equation is quite straightforward because 7 is already less than 10 and we can quickly see what x needs to be added to 7 to make it leave a remainder of 5 when divided by 10.
The equation asks for x where 7 + x ≡ 5 mod 10. Since 7 + 3 = 10, and 10 leaves a remainder of 0 when divided by 10, we need to go 5 steps further to get a remainder of 5. Thus, 7 + 8 = 15 is the smallest positive solution since 15 divided by 10 leaves a remainder of 5.
So, x = 8 is the correct solution. It's always important to check the answer to ensure it's reasonable. In this case, we can see that 15 does indeed leave a remainder of 5 when divided by 10, confirming that our solution is reasonable and x = 8 is the smallest positive solution as required.