Is my solution correct?

[untold sparrow]

why decompose 653 as 600 and 53? try dividing out fully by 3

[limber sparrow]

but it's not divisible by 3

[untold sparrow]

yes, it will have a remainder

the idea is that if you play around with the function it is cyclic

so you take advantage of that

[limber sparrow]

Thats what i did

I first calculated f(x)^1

then f(x)^2 f(x)^3 and so on

it repeats at the 3rd power

[untold sparrow]

great!

so that motivates you to divide it by 3

[limber sparrow]

I don't understand

[untold sparrow]

basically, the exponent, when written in the form 3z+r, will reduce to just r

because of the cyclic nature

that makes sense?

[limber sparrow]

sorry no

[untold sparrow]

for example, notice that f_4 will reduce to f_1

because 4 = 3(1) + 1

and so on and so forth

[limber sparrow]

4 = 3(1) + 1 why (1) + 1?

[untold sparrow]

thats the result of dividing 4 by 3

3 goes into 4 once and remainder is 1

It seems like you have the big picture just need to clean up the details

So 653 = 3(217) + 2, and so f_653 = f_2

and since you've already done the work with f_2, its breezy from there

are you still lost?

[limber sparrow]

That is cool. I didn't know about that method

It is clear now. Thank you

[untold sparrow]

im curious, where is this problem from?

[limber sparrow]

Its from the book "How to think like a Mathematician" by Kevin Houston Chapter 5

[untold sparrow]

for a intro to proofs class i assume?

[limber sparrow]

No I am self studying to prepare for University

[untold sparrow]

keep up the work!

[limber sparrow]

Thanks

.solved