Prove that if a and b are integers, then for any integer k one has (a, b) = (a + kb, b). (Hint: Show that they are mutually divisible.)

The operation:

(a,b) is equal to the rest of the division of a by b.

Now, if we have:

(a + kb,b) = (a,b) + (k*b,b)

But if we have that k and b are integers, then:

(k*b)/b = k

So b divides k*b into a whole number, this means that (k*b,b) = 0

then:

(a + kb,b) = (a,b) + (k*b,b) = (a,b) + 0 = (a,b)

the entire expression should be subtracted from 30. to subtract the quantity, distribute the negative sign, which changes the signs on the terms. so, the –2 changes to positive and gets added to 30 instead of subtracted from it.

hope this ! ^w^

equal to

step-by-step explanation: