Question 24 of 26intermediate🔧 ApplyLong Answer3 marks

Use proof by contradiction to prove that if for an integer aa, a2a^2 is divisible by 3, then aa is divisible by 3.

Correct Answer

Let us assume that the statement is not true. This means we assume that a is not divisible by 3.

Since aa is an integer not divisible by 3, using Euclid's division lemma, it must be of the form 3q+13q+1 or 3q+23q+2 for some integer qq.

Case 1: If a=3q+1a = 3q+1, then

a2=9q2+6q+1=3(3q2+2q)+1a^2 = 9q^2 + 6q + 1 = 3(3q^2 + 2q) + 1

which is not divisible by 3.

Case 2: If a=3q+2a = 3q+2, then

a2=9q2+12q+4=3(3q2+4q+1)+1a^2 = 9q^2 + 12q + 4 = 3(3q^2 + 4q + 1) + 1

which is also not divisible by 3. In both cases, a2a^2 is not divisible by 3. This contradicts the hypothesis that a2a^2 is divisible by 3. The contradiction has arisen because we assumed that aa is not divisible by 3.

∴ aa must be divisible by 3.

Exercise: EXERCISE A1.6 | Q: 4 | (Page 21)
For More Understanding

Explanation

This answer follows the structure of 'proof by contradiction' outlined in the provided text (specifically section A1.7 and the table). It starts by assuming the negation of the statement, uses Euclid's division lemma (mentioned in the context table) to explore cases, derives a contradiction to the hypothesis, and concludes the original statement is true.

Solution Steps

  1. Step 1: Assume the statement is not true, i.e., aa is not divisible by 3.

  2. Step 2: Express aa as 3q+13q+1 or 3q+23q+2 using Euclid's division lemma.

  3. Step 3: Square these expressions to show a2a^2 leaves a remainder of 1, meaning a2a^2 is not divisible by 3.

  4. Step 4: Identify the contradiction with the given condition that a2a^2 is divisible by 3.

  5. Step 5: Conclude that the assumption is false, so aa must be divisible by 3.

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