cramya wrote:Hi Ian,
Thank you!
Question:
If individually x^16, y^8 are both divisible by 15 then the combination x^16-y^8+345y^2 is divisible by 15. (someone correct me here if I am mistaken)
Is this correct or am I mistaken?
Also lets say for some reason we find y^8 to be never be divisible by 15 can we for sure say the expression will not be divisible by 15?
What are the norms/rules when there are individual components like these in an expression and the question of divisibility by a number comes in to play?
Please advice.
If you add two (or more) multiples of 15, you'll always get a multiple of 15. Say x and y are multiples of 15. Then we can write:
x = 15a for some integer a
y = 15b for some integer b
so x+y = 15a + 15b = 15(a+b), and we see that x+y is a multiple of 15.
On the other hand, say x is a multiple of 15, and z is not - say z gives a remainder of r when we divide by 15. Then we can write:
x = 15a
z = 15q + r
so x+z = 15a + 15q + r = 15(a+q) + r, and we see that we will get a remainder of r when we divide x+z by 15.
What if we add two numbers, neither of which is a multiple of 15? Here you'd need more information to say anything. We can, as above, write our numbers:
w = 15Q + R
z = 15q + r
and w+z = 15(Q + q) + R + r, but here, R+r might be a multiple of 15, so it's possible that w+z is a multiple of 15, possible that it's not.
And of course you can replace 15 throughout the above with any positive integer 2 or larger.
