Number Play | Part 3 | Class 8 Maths | Chapter 5 | Ganita Prakash

Summary

The lesson explains parity—whether a number is even or odd—and uses it to determine whether numerical or algebraic expressions are always even without calculating their exact values. It also investigates when the sum of two even numbers is divisible by four.

Rules for Even and Odd Numbers

• Adding or subtracting two even numbers gives an even number.
• Adding or subtracting two odd numbers gives an even number.
• Adding or subtracting one even and one odd number gives an odd number.
• Multiplying an even number by any integer gives an even number.
• Multiplying two odd numbers gives an odd number.
• A number’s parity can be identified from its last digit:
  • 0, 2, 4, 6, or 8: even
  • 1, 3, 5, 7, or 9: odd
• Negative numbers can also be even or odd; the same rules apply.

These rules let students assess sums, differences, and products without working out their full values.

Applying Parity to Algebraic Expressions

The teacher extends the rules to expressions involving variables, assuming the variables represent integers.

• An expression with an even factor, such as 2a or 4m, is always even, regardless of the variable’s value.
• An expression such as 3g is not necessarily even; its value depends on whether g is even or odd.
• Expressions with the same variable can sometimes be simplified first. For example, 13k − 5k = 8k, which is always even because 8 is even.
• In 6m − 3n, the first term is always even, but the parity of the second depends on n. Therefore, the whole expression is not always even.
• x² + 2 and b² + 1 are also not always even; their parity depends on whether the variable is even or odd.
• A product with an even factor, such as 4k × 3j, is always even.

To decide whether an expression is always even:

1. Identify which parts are certainly even or odd.
2. Check whether any variable-dependent part can change parity.
3. Simplify terms with the same variable when possible.
4. Conclude that the expression is “always even” only if the rules guarantee an even result for every integer value of the variables.

Expressions built from multiples of 2—for example, sums or differences of terms such as 2a, 4b, and 6c—are always even.

When Does a Pair of Even Numbers Add to a Multiple of Four?

An even number is either a multiple of four, such as 4, 8, 12, or 16, or an even number that is not a multiple of four, such as 2, 6, 10, or 14.

When an even number is divided by 4, its remainder is always either 0 or 2. This gives three cases:

1. Both numbers are multiples of four: Their sum is a multiple of four. 4m + 4n = 4(m + n)

2. Neither number is a multiple of four: Each has the form 4m + 2, so their sum is a multiple of four. (4m + 2) + (4n + 2) = 4(m + n + 1)

3. One number is a multiple of four and the other is not: Their sum is not a multiple of four. 4m + (4n + 2) = 4(m + n) + 2

Thus, the sum of two even numbers is divisible by four when both are multiples of four or when neither is. It is not divisible by four when exactly one is.

The teacher uses general forms to show that these conclusions hold for all such numbers, rather than relying only on a few examples.

Speakers and Sources

• Speaker: One instructor, addressing the class. No students or other speakers are heard.
• Channel/source: Learn With Mansi Juniors.