Question

(1,3,5,7,9,11,13,15)
......+......+...... = 30 by using these numbers

Answer 1

Answer:

Can never be formed!!!

Concept used:

The sum of any three positive odd integers is always odd.

Reason:

Given are of odd numbers from 1 to 15.

And the question is to write 30 as sum of three numbers using positive odd integers from 1 to 15.

As the concept given above, the sum of the numbers which have to be used in the LHS is odd while the RHS 30 is even!!!

Thus a contradiction occurs here.

Algebraic proof:

As odd numbers leave remainder 1 on division by 2, let the odd integers which have to be used in the LHS to get the sum 30 be 2x + 1, 2y + 1 and 2z + 1, for any distinct whole numbers from 0 to 7   x, y, z.

Thus,

⇒ 2x + 1 + 2y + 1 + 2z + 1 = 30

⇒ 2x + 2y + 2z + 3 = 30

⇒ 2(x + y + z) + 3 = 30

⇒ 2(x + y + z) = 30 - 3

⇒ 2(x + y + z) = 27

Here, what the contradiction occurred is that the LHS is even while the RHS is odd!

After 2(x + y + z) = 27,

⇒ x + y + z = 27 / 2

⇒ x + y + z = 13.5

Here, is it true that 13.5 can be written as sum of three whole numbers?!