Proofs (1 Viewer)

[kractus]

How would you do both these questions? I don't know where to start

 

[011235]

Q14 (a)

Let n=2k+1 (odd) where k is a positive integer.

Let n consecutive numbers be a+1, a+2, a+3, ..., a+n for some integer a.

The sum of these is;



Hence the sum is divisible by n.

Now try using the same logic for (b)

 

[jimmysmith560]

Would the following working help with Question 15?

Let the digits of a 4-digit number n be a, b, c, d.

That is,







Part 1:

If n is divisible by 3, then for some integer m.

Therefore:





Therefore the sum of the digits (𝑎 + 𝑏 + 𝑐 + 𝑑) is a multiple of 3.

Part 2:

If the sum of the digits is a multiple of 3, then (𝑎 + 𝑏 + 𝑐 + 𝑑) = 3𝑘 for some integer 𝑘.

Therefore:





Therefore 𝑛 is divisible by 3. We have now proved the result in both directions, so a 4-digit number is divisible by 3 if and only if the sum of its digits is divisible by 3.