Question 15 of 25intermediate💡 UnderstandShort Answer3 marks

Take a pair of even numbers. Add them. Is the sum divisible by 4?

Correct Answer

The sum of two even numbers may or may not be divisible by 4.

Case 1: Both even numbers are multiples of 4 (like 4 and 8) Their sum is divisible by 4.

Case 2: Both even numbers leave remainder 2 when divided by 4 (like 6 and 10) Their sum is divisible by 4.

Case 3: One is a multiple of 4 and the other leaves remainder 2 (like 4 and 6) Their sum is NOT divisible by 4.

Mathematical Reasoning: Even numbers divisible by 4 have form 4p. Even numbers not divisible by 4 have form 4p + 2. Adding two multiples of 4: 4p + 4q = 4(p+q) ∴ divisible by 4. Adding two numbers of form 4p+2: (4p+2) + (4q+2) = 4(p+q+1) ∴ divisible by 4. Adding a multiple of 4 and a number of form 4p+2: 4p + (4q+2) = 4(p+q) + 2 ∴ NOT divisible by 4.

Exercise: Pairs to Make Fours | Q: 1 | (Page Page 5)
For More Understanding

Explanation

The chapter explains that even numbers can be classified by their remainder when divided by 4. Even numbers divisible by 4 have the form 4p. Even numbers not divisible by 4 have the form 4p + 2. Adding two multiples of 4: 4p + 4q = 4(p+q), divisible by 4. Adding two numbers of form 4p+2: (4p+2) + (4q+2) = 4(p+q+1), divisible by 4. Adding a multiple of 4 and a number of form 4p+2: 4p + (4q+2) = 4(p+q) + 2, NOT divisible by 4.

Solution Steps

  1. Classify even numbers by their remainder when divided by 4: multiples of 4 (form 4p) and numbers leaving remainder 2 (form 4p+2).

  2. Case 1: Both numbers are multiples of 4 → sum is divisible by 4.

  3. Case 2: Both numbers leave remainder 2 → sum is divisible by 4.

  4. Case 3: One multiple of 4, one remainder 2 → sum is NOT divisible by 4.

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