Need guidance regarding Remainder & exponents. Till remainder deux ,it was easy but after that I have noticed that the rules are different when it comes to remainder & exponents

Hi can you clarify further?

When to do consider unit digit and directly divide the numerator by denominator?? I have found that Greg in some maths did the divination and in some maths considered exponents

Just to clarify, weâ€™re finding the remainder in all these cases - there is no real â€śdivisionâ€ť going on. So for instance, in the last screenshot, weâ€™re finding the remainder when 36^{17} is divided by 2.

That means you want to say I have to find unit digit, unit digit of 6 is 6 which will divide the exponent, the remainder will be the answer or again have to find which digit has that remainder as we did in remainder chapters? I donâ€™t know whether I have made it clear or not, but the chapter is making me freak out

The question is asking what the remainder is, so thatâ€™s the answerâ€¦

Keep in mind that â€śunit digitâ€ť is not the same as â€śremainderâ€ť, except when youâ€™re dividing the number by 10.

Just tell me I just have to divide the numerator with denominator & the remainder would be the answer? Exponent has nothing to do with it?

Exponent definitely has something to do with it. I think youâ€™re missing the point.

Letâ€™s try this: what is the remainder when say 5^{49} is divided by 3? Show your steps clearly.

Remainder is 2? After that what to do?

How did you get that?

5/3

By your logic, the remainder of 5^x (where x is a positive integer) should be 2 when divided by 3. Is that right?

Yea

And thatâ€™s where your logic fails, because that isnâ€™t correct. The whole point of these kind of problems is to *find a pattern*.

Letâ€™s try this with the previous example:

- The remainder when 5 is divided by 3 is 2
- The remainder when 5^2 = 25 is divided by 3 is 1
- The remainder when 5^3 = 125 is divided by 3 is 2
- The remainder when 5^4 = 625 is divided by 3 is 1

Do you see a pattern?

Oh that means in this math we have to at first divide the exponents of unit digit by the denominator in question?? Then what about 49?

No. Weâ€™re finding the pattern by explicitly calculating 5^1, 5^2, 5^3, â€¦ and finding the remainders for these (small) numbers - we can use the calculator for that after all. Then we use that to find a pattern we can use to solve the problem for larger powers of 5.

In my previous answer, can you notice that when x is even, the remainder of 5^x when divided by 3 is 1? What if x is odd? Can you use that to answer the question on the remainder of 5^{49} when divided by 3?

No, because exponent is 49, so answer would be 2??

Well yes, because 49 is odd, and weâ€™ve established via the pattern that when x is odd, the remainder would be 2.

Can you now apply this to the original problem you gave in the question?

Thanks for clarification, U did it step by step thatâ€™s why, I understood, thanks a lot