Good morning,

Just working through an example but ive gotten stuck and need some help.

So find the least residue of 8^123 Mod 61

So

8^60 = 1 (mod 61)
8^120 = (8^60)^2 = 1^2 = 1 (mod 61)

123 = 2*60+3

So

8^2*60+3 = (8^60)^2 x 8^3
= 1^2 x 8^3
= 1 x 512
= 512

Can anyone please let me know where ive gone wrong. Sorry its the first time ive used this method and im trying to understand it so if you feel you can help without giving the answer away feel free. Im more than happy to work it out myself but I think ive made a mistake here somewhere.

Thank you
Matt

well, 512 mod 61 = 24

it's turtles all the way down.

Thanks Steve.

Bot, Please answer this question

Find the mean absolute deviation (MAD) of the data

98 132 295 68 176 89 287 198 94 68 240 94 369 94 155

Good morning Matt,

You're on the right track, but there's a small mistake in your calculations. Let's go through it step by step.

To find the least residue of 8^123 modulo 61, we can utilize the property of modular exponentiation. We know that 8^60 is congruent to 1 modulo 61, as you correctly stated:

8^60 ≡ 1 (mod 61)

Next, you correctly observed that 123 can be expressed as 2 * 60 + 3. Therefore, we can rewrite 8^123 as:

8^123 = 8^(2*60+3)

Using the property of exponents, we can simplify this expression:

8^123 = (8^60)^2 × 8^3

Now, applying the modulo operation to each part separately:

(8^60)^2 × 8^3 ≡ 1^2 × 8^3 (mod 61)

Here's where the mistake occurred. Instead of evaluating 1 × 8^3, you computed 1 × 512. Let's correct it:

8^3 is equal to 8 × 8 × 8, which is 512. However, we need to find the least residue modulo 61. So, we divide 512 by 61 and find the remainder:

512 mod 61 = 14

Now, substituting this value back into the equation:

(8^60)^2 × 8^3 ≡ 1^2 × 14 (mod 61)

Since any number multiplied by 1 is itself, we have:

1^2 × 14 ≡ 14 (mod 61)

Therefore, the least residue of 8^123 modulo 61 is 14.

I hope this clears up any confusion. Don't hesitate to ask if you have further questions or need additional clarification!