Find the least residue of 7^5 mod 50 without using a calculator.
so far I have 7=7 mod 50
7^5 = 7^5 mod 50
7^2 = 49
7^3=343
since 7^2=(-1)mod50
and 7^3=(-43)mod50
it follows that
7^6=7^2 * 7^3
= (-1)(-43)
= 43
feel like this is wrong though could somebody please explain where I have made an error?
To find the least residue of 7^5 mod 50 without using a calculator, you need to apply the concept of modular arithmetic. Let's break down your steps and identify any errors:
1. Start with the statement: 7^5 ≡ 7^5 (mod 50). This part is correct as it just sets up the equation.
2. In your next step, you calculated 7^2 ≡ 49 (mod 50). Again, this is correct.
3. However, in the subsequent step, you stated that 7^3 ≡ -43 (mod 50). This is where the error lies. The correct calculation should be:
7^3 ≡ (7^2)(7) ≡ (49)(7) ≡ 343 (mod 50).
Now, it's important to simplify this result by taking the remainder when dividing 343 by 50:
343 ≡ 343 ÷ 50 ≡ 6 remainder 43 ≡ 43 (mod 50).
4. Now, let's use this corrected result and proceed to the final step you attempted:
7^5 = 7^2 * 7^3 ≡ 49 * 43 (mod 50).
Again, we simplify by taking remainders:
49 * 43 ≡ 26 * 43 ≡ 1118 ≡ 18 remainder 18 ≡ 18 (mod 50).
So, the correct answer is that the least residue of 7^5 mod 50 is 18.