Practice Direct Proof in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A direct proof establishes a statement P⇒QP \Rightarrow Q by assuming PP is true and using logical steps, definitions, and known theorems to arrive at QQ — the most straightforward proof strategy.

Start from what you know (the hypotheses) and chain logical steps forward until you reach what you want to prove — no detours, no tricks, just forward reasoning.

Showing a random 20 of 50 problems.

Example 1

hard
Direct proof: if nn is an odd integer, then n2≡1(mod8)n^2 \equiv 1 \pmod 8.

Example 2

medium
Direct proof that if n=2k+1n=2k+1 then n2−1n^2-1 is divisible by 88. Complete it.

Example 3

easy
Finish: 'If n=4kn = 4k, then n=2(2k)n = 2(2k), so nn is ____.'

Example 4

easy
Direct proof of 'if aa and bb are positive then ab>0ab > 0': what justifies the conclusion?

Example 5

easy
Direct proof that the product of two odd integers is odd: assume a=2j+1a=2j+1, b=2k+1b=2k+1. Expand abab.

Example 6

medium
Complete the direct proof that nn odd ⇒n2\Rightarrow n^2 odd: assume n=2k+1n=2k+1, then n2=n^2 = ? Show it is odd.

Example 7

medium
Prove directly: for all real xx, (x−1)(x+1)=x2−1(x-1)(x+1) = x^2 - 1.

Example 8

hard
Direct proof: if aa is rational and bb is irrational, must a+ba + b be irrational?

Example 9

medium
Direct proof: if aa and bb are consecutive integers then a+ba+b is odd. Complete it.

Example 10

medium
Direct proof: if xx is rational and yy is rational, then xyxy is rational. Complete it.

Example 11

easy
In a direct proof of 'nn is a multiple of 4 ⇒n\Rightarrow n is even', what is the opening assumption?

Example 12

medium
Direct proof: if aa and bb are integers with a≡b(modm)a \equiv b \pmod{m}, then a2≡b2(modm)a^2 \equiv b^2 \pmod{m}.

Example 13

hard
Prove directly: for all positive reals a,ba, b, a2+b2≥2aba^2 + b^2 \ge 2ab.

Example 14

medium
Direct proof: if a∣ba \mid b and b∣cb \mid c then a∣ca \mid c. Set up the definitions and finish.

Example 15

medium
In direct proofs, what is the role of the definition of the hypothesis (e.g., 'even', 'rational')?

Example 16

easy
Direct proof that 3∣n⇒3∣n23 \mid n \Rightarrow 3 \mid n^2: assume n=3kn=3k. What is n2n^2?

Example 17

easy
In a direct proof of P⇒QP \Rightarrow Q, can we begin by assuming ¬Q\neg Q?

Example 18

easy
In a direct proof of 'if nn is even then n2n^2 is even', what do you assume FIRST?

Example 19

medium
Direct proof: for all reals a,ba, b, (a+b)2≥4ab(a+b)^2 \ge 4ab. Hint: expand.

Example 20

medium
Direct proof that if nn is odd, then n2−1n^2 - 1 is divisible by 4. Complete it.
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