Number Play | Class 8 Mathematics Notes

Pairs to Make Fours

This section investigates the conditions under which the sum of two even numbers is divisible by 4. Students experiment by adding pairs of even numbers and observing whether their sum is a multiple of 4. Through exploration, they discover that even numbers can be categorized based on their remainder when divided by 4: those that are multiples of 4 (remainder 0) and those that are not (remainder 2).

Using algebra, the section explains three cases:

1. Sum of two even numbers both multiples of 4: 4p + 4q = 4(p + q), always a multiple of 4.

2. Sum of two even numbers both leaving remainder 2 when divided by 4: (4p + 2) + (4q + 2) = 4(p + q + 1), also a multiple of 4.

3. Sum of one multiple of 4 and one even number leaving remainder 2: 4p + (4q + 2) = 4(p + q) + 2, which is not a multiple of 4.

Visualizations accompany these algebraic explanations to aid understanding. The section encourages students to identify patterns, generalize rules, and understand the interplay between remainders and divisibility.

This exploration deepens students' grasp of modular arithmetic concepts and prepares them for more complex divisibility properties.

📊 Diagram: See figure_8: Even numbers that are multiples of 4 leave a remainder of 0 when divided by 4; See figure_9: Even numbers that are not multiples of 4 leave a remainder 2 when divided by 4.

🧪 Activity: Students add pairs of even numbers, classify them by their remainder modulo 4, and verify algebraic explanations with visual models.

🔗 Connection: This section's understanding of remainders and divisibility leads into 'Always, Sometimes, or Never', which examines the truth of statements about divisibility and multiples.