- #1
spaghetti3451
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Homework Statement
(a) Prove that if n is an integer and n2 is a multiple of 3, then n is a multiple of 3.
(b) Consider a class of n students. In an exam, the class average is k points. Prove, using contradiction, that at least one student must have received at least k marks in the exam.
Homework Equations
The Attempt at a Solution
(a) I think I've solved the first one correctly. Here's my attempt.
Assume that n is not a multiple of 3.
Therefore, n ≠ 3p, where p is an integer.
Therefore, n2 ≠ 9p2.
Therefore, n2 ≠ 3(3p2).
Therefore, n2 is not a multiple of 3.
Therefore, by contraposition, if n2 is a multiple of 3, then n is a multiple of 3.
(b) The second one I've made a partial attempt as I could not figure how to proceed. Here's my attempt.
Assume that class average ≠ k points.
Therefore, total sum of marks of all students ≠ nk points.
No idea of how to proceed from here onwards.
It would be great if you could supply hints on how to carry forward and check my first solution as well.