Lifting-the-exponent lemma
The lifting-the-exponent lemma is like when you have a big number and you want to make it bigger by multiplying it by itself a bunch of times.
Here's an example: let's say you have the number 2, and you want to make it bigger by multiplying it by itself 5 times (2 to the power of 5).
But then maybe you want to see how many times 2 divides into that big number you just made (2 to the power of 5).
The lifting-the-exponent lemma helps you figure that out by looking at how many times 2 divides into the original number (2 to the power of 1), and then seeing how many times 2 divides into the difference between the big number and the original number.
Basically, it helps you figure out how many times a small number (like 2) divides into a big number that you made by multiplying that small number by itself a bunch of times.
Here's an example: let's say you have the number 2, and you want to make it bigger by multiplying it by itself 5 times (2 to the power of 5).
But then maybe you want to see how many times 2 divides into that big number you just made (2 to the power of 5).
The lifting-the-exponent lemma helps you figure that out by looking at how many times 2 divides into the original number (2 to the power of 1), and then seeing how many times 2 divides into the difference between the big number and the original number.
Basically, it helps you figure out how many times a small number (like 2) divides into a big number that you made by multiplying that small number by itself a bunch of times.