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Number Play - Picking Parity

Grade 7CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Parity refers to whether an integer is even or odd. An even number is any integer that can be divided by 22 with no remainder, represented as 2n2n. An odd number is an integer that leaves a remainder of 11 when divided by 22, represented as 2n+12n + 1.

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The sum of any number of even integers is always even because 2n+2m=2(n+m)2n + 2m = 2(n + m).

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The sum of two odd integers is always even. For example, (2n+1)+(2m+1)=2(n+m+1)(2n + 1) + (2m + 1) = 2(n + m + 1), which is a multiple of 22.

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A product of integers is even if at least one factor in the product is even. If all factors are odd, the product is odd.

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For any integer xx, the parity of xnx^n (where n>0n > 0) is the same as the parity of xx. For example, 353^5 is odd because 33 is odd, and 4104^{10} is even because 44 is even.

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The sum of kk odd numbers is even if kk is even, and odd if kk is odd.

📐Formulae

Even±Even=EvenEven \pm Even = Even

Odd±Odd=EvenOdd \pm Odd = Even

Even±Odd=OddEven \pm Odd = Odd

Even×Even=EvenEven \times Even = Even

Odd×Odd=OddOdd \times Odd = Odd

Even×Odd=EvenEven \times Odd = Even

∑i=1n(2ki)=2∑ki (Sum of evens is even)\sum_{i=1}^{n} (2k_i) = 2\sum k_i \text{ (Sum of evens is even)}

💡Examples

Problem 1:

Determine if the result of the following expression is even or odd without calculating the full value: 1234+5679+88821234 + 5679 + 8882.

Solution:

The expression consists of: Even(1234)+Odd(5679)+Even(8882)Even (1234) + Odd (5679) + Even (8882).

Explanation:

Using parity rules: Even+Odd=OddEven + Odd = Odd. Then, Odd+Even=OddOdd + Even = Odd. Therefore, the final result is Odd.

Problem 2:

Calculate the sum of the first four odd numbers and check its parity using vertical addition: 1,3,5,71, 3, 5, 7.

Solution:

135+716\begin{array}{r} 1 \\ 3 \\ 5 \\ + 7 \\ \hline 16 \end{array}

Explanation:

The sum of 4 (an even number) odd numbers should be even. 1+3+5+7=161+3+5+7 = 16. Since 1616 ends in 66, it is an even number, confirming the rule.

Problem 3:

Is the product 21×35×47×99×221 \times 35 \times 47 \times 99 \times 2 even or odd?

Solution:

The product is Even.

Explanation:

In a multiplication string, if even one number is even, the entire product becomes even. Here, the factor 22 is even, so Odd×Odd×Odd×Odd×Even=EvenOdd \times Odd \times Odd \times Odd \times Even = Even.

Problem 4:

What is the parity of 1520+121515^{20} + 12^{15}?

Solution:

152015^{20} is Odd and 121512^{15} is Even. Odd+Even=OddOdd + Even = Odd.

Explanation:

Since 1515 is odd, any positive power of 1515 is odd (Oddn=OddOdd^n = Odd). Since 1212 is even, any positive power of 1212 is even (Evenn=EvenEven^n = Even). The sum of an odd and an even number is always odd.