Context: Consider the algebraic expression: . For different values of , the expression has different parity: (a) Come up with an expression that always has even parity. Some examples are: and . Try to find more. (b) Come up with expressions that always have odd parity. (c) Come up with other expressions, like , which could have either odd or even parity. (d) The expression evaluates to (for ) — many even numbers are missing. (e) Are there expressions using which we can list all the even numbers? Hint: All even numbers have a factor . (f) Are there expressions using which we can list all odd numbers?

Understand the Question Parity describes whether an integer is even or odd. Even numbers are divisible by and can always be written in the form (where is an integer). Odd numbers leave a remainder of when divided by and can always be written in the form (or ). Some algebraic expressions always produce even numbers, some always produce odd numbers, and others vary in parity depending on the value of the variable. Part (a) Come up with an expression that always has even parity. Some examples are: and . Try to find more. Step 1 · Expressions with Even Parity An even number is always divisible by and can be factored as . For any integer : Answer > (a) , , , Part (b) Come up with expressions that always have odd parity. Step 1 · Expressions with Odd Parity An odd number is not divisible by and leaves a remainder of when divided by , so it can be expressed in the form . For any integer : Answer > (b) , , , Part (c) Come up with other expressions, like , which could have either odd or even parity. Step 1 · Expressions with Mixed Parity An expression has mixed parity when its value alternates between even and odd depending on the variable. For : If (even) (odd) If (odd) (even) For : If (odd) (odd) If (even) (even) Answer > (c) , Part (d) The expression evaluates to (for ) — many even numbers are missing. Step 1 · Evaluate and Identify Missing Numbers Evaluating for : All obtained values are even, but the expression only generates even numbers separated by . It misses many even numbers, such as Answer > (d) The expression only generates a subset of even numbers and misses even numbers such as Part (e) Are there expressions using which we can list all the even numbers? Hint: All even numbers have a factor . Step 1 · Expression for All Even Numbers All even numbers are multiples of . For any integer : Hence, covers all even numbers. Answer > (e) Yes, the expression (where is any integer) lists all even numbers. Part (f) Are there expressions using which we can list all odd numbers? Step 1 · Expression for All Odd Numbers Every odd number is more (or less) than an even number. For any integer : Hence, (or ) covers all odd numbers. Answer > (f) Yes, the expression (or ), where is any integer, lists all odd numbers. Common Mistakes Assuming Any Even Multiplier Generates All Evens: Expressions like or generate only a subset of even numbers. Only produces every even integer. Ignoring Zero and Negative Integers: Parity applies to all integers: are even, and are odd. Confusing the Multiplier with the Constant: In an expression like , the output parity varies because the multiplier of is odd (), not because of the even constant .

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Question 13

Context: Consider the algebraic expression: $3n + 4$ . For different values of $n$ , the expression has different parity:

(a) Come up with an expression that always has even parity. Some examples are: $100p$ and $48w - 2$ . Try to find more.

(b) Come up with expressions that always have odd parity.

(d) The expression $6k + 2$ evaluates to $8, 14, 20, \dots$ (for $k = 1, 2, 3, \dots$ ) — many even numbers are missing.

(e) Are there expressions using which we can list all the even numbers? Hint: All even numbers have a factor $2$ .

(f) Are there expressions using which we can list all odd numbers?

Solution

Understand the Question

Parity describes whether an integer is even or odd.
Even numbers are divisible by $2$ and can always be written in the form $2k$ (where $k$ is an integer).
Odd numbers leave a remainder of $1$ when divided by $2$ and can always be written in the form $2k + 1$ (or $2k - 1$ ).
Some algebraic expressions always produce even numbers, some always produce odd numbers, and others vary in parity depending on the value of the variable.

(a) Come up with an expression that always has even parity. Some examples are: $100p$ and $48w - 2$ . Try to find more.

Step 1 · Expressions with Even Parity

An even number is always divisible by $2$ and can be factored as $2 \times (\text{integer})$ . Diagram 1

For any integer $x$ :

$2x$
$4x = 2(2x)$
$2x + 6 = 2(x + 3)$
$10x - 4 = 2(5x - 2)$

Answer

(a) $2x$ , $4x$ , $2x + 6$ , $10x - 4$

(b) Come up with expressions that always have odd parity.

Step 1 · Expressions with Odd Parity

An odd number is not divisible by $2$ and leaves a remainder of $1$ when divided by $2$ , so it can be expressed in the form $2 \times (\text{integer}) + 1$ .

For any integer $y$ :

$2y + 1$
$4y + 1 = 2(2y) + 1$
$2y - 3 = 2(y - 2) + 1$
$6y + 5 = 2(3y + 2) + 1$

Answer

(b) $2y + 1$ , $4y + 1$ , $2y - 3$ , $6y + 5$

(c) Come up with other expressions, like $3n + 4$ , which could have either odd or even parity.

Step 1 · Expressions with Mixed Parity

An expression has mixed parity when its value alternates between even and odd depending on the variable.

For $n + 5$ :
- If $n = 2$ (even) $\implies n + 5 = 7$ (odd)
- If $n = 3$ (odd) $\implies n + 5 = 8$ (even)
For $5n + 2$ :
- If $n = 1$ (odd) $\implies 5n + 2 = 7$ (odd)
- If $n = 2$ (even) $\implies 5n + 2 = 12$ (even)

Answer

(c) $n + 5$ , $5n + 2$

(d) The expression $6k + 2$ evaluates to $8, 14, 20, \dots$ (for $k = 1, 2, 3, \dots$ ) — many even numbers are missing.

Step 1 · Evaluate $6k + 2$ and Identify Missing Numbers

Evaluating $6k + 2$ for $k = 1, 2, 3$ :

\begin{aligned} 6(1) + 2 &= 8 \\ 6(2) + 2 &= 14 \\ 6(3) + 2 &= 20 \end{aligned}

All obtained values are even, but the expression only generates even numbers separated by $6$ . It misses many even numbers, such as $2, 4, 6, 10, 12, 16, 18, \dots$

Answer

(d) The expression $6k + 2$ only generates a subset of even numbers and misses even numbers such as $2, 4, 6, 10, 12, \dots$

(e) Are there expressions using which we can list all the even numbers? Hint: All even numbers have a factor $2$ .

Step 1 · Expression for All Even Numbers

All even numbers are multiples of $2$ .

For any integer $m \in \{0, \pm 1, \pm 2, \dots\}$ :

$m = 0 \implies 2(0) = 0$
$m = 1, -1 \implies 2(1) = 2, \; 2(-1) = -2$
$m = 2, -2 \implies 2(2) = 4, \; 2(-2) = -4$

Hence, $2m$ covers all even numbers.

Answer

(e) Yes, the expression $2m$ (where $m$ is any integer) lists all even numbers.

(f) Are there expressions using which we can list all odd numbers?

Step 1 · Expression for All Odd Numbers

Every odd number is $1$ more (or $1$ less) than an even number.

For any integer $m \in \{0, \pm 1, \pm 2, \dots\}$ :

$m = 0 \implies 2(0) + 1 = 1$
$m = 1, -1 \implies 2(1) + 1 = 3, \; 2(-1) + 1 = -1$
$m = 2, -2 \implies 2(2) + 1 = 5, \; 2(-2) + 1 = -3$

Hence, $2m + 1$ (or $2m - 1$ ) covers all odd numbers.

Answer

(f) Yes, the expression $2m + 1$ (or $2m - 1$ ), where $m$ is any integer, lists all odd numbers.

Common Mistakes

Assuming Any Even Multiplier Generates All Evens: Expressions like $4m$ or $6k + 2$ generate only a subset of even numbers. Only $2m$ produces every even integer.
Ignoring Zero and Negative Integers: Parity applies to all integers: $0, -2, -4$ are even, and $-1, -3$ are odd.
Confusing the Multiplier with the Constant: In an expression like $3n + 4$ , the output parity varies because the multiplier of $n$ is odd ( $3$ ), not because of the even constant $+4$ .

More questions in IT

What do the numbers in the figure below tell us?

Remember the children from the Grade 6 textbook of mathematics? Now, they call out numbers using a different rule.

What do you think these numbers mean?

The children rearrange themselves and each one says a number based on the new arrangement.

Context: The children rearrange themselves and each one says a number based on the new arrangement.

Q. Could you figure out what these numbers convey? Observe and try to find out.

Write down the number each child should say based on this rule for the arrangement shown below.

Kishor has some number cards and is working on a puzzle: There are 5 boxes, and each box should contain exactly 1 number card. The numbers in the boxes should sum to 30. Can you help him find a way to do it?

Can you figure out which 5 cards add to 30? Is it possible? There are many ways of choosing 5 cards from this collection. Is there a way to find a solution without checking all possibilities? Let us find out.

Context: Kishor has some number cards and is working on a puzzle: There are 5 boxes, and each box should contain exactly 1 number card. The numbers in the boxes should sum to 30. Can you help him find a way to do it? Can you figure out which 5 cards add to 30? Is it possible? There are many ways of choosing 5 cards from this collection. Is there a way to find a solution without checking all possibilities? Let us find out.

Q. Add a few even numbers together. What kind of number do you get? Does it matter how many numbers are added?

Context: As we see in the figure, adding any number of even numbers will result in a number which can still be arranged in pairs without any leftovers. In other words, the sum will always be an even number.

Q. Now, add a few odd numbers together. What kind of number do you get? Does it matter how many odd numbers are added?

Context: Can we also think of an odd number as one less than a collection of pairs? This figure shows that the sum of two odd numbers must always be even! This along with the other figures here are more examples of a proof!

Q. What about adding 3 odd numbers? Can the resulting sum be arranged in pairs?

Explore what happens to the sum of:

(a) 4 odd numbers

(b) 5 odd numbers

Q10

Two siblings, Martin and Maria, were born exactly one year apart. Today they are celebrating their birthday. Maria exclaims that the sum of their ages is 112. Is this possible? Why or why not?

Q11

Context: Small Squares in Grids In a $3 \times 3$ grid, there are 9 small squares, which is an odd number. Meanwhile, in a $3 \times 4$ grid, there are 12 small squares, which is an even number.

Q. Given the dimensions of a grid, can you tell the parity of the number of small squares without calculating the product?

Q12

Find the parity of the number of small squares in these grids:

(a) $27 \times 13$

(b) $42 \times 78$

Q13

Context: Consider the algebraic expression: $3n + 4$ . For different values of $n$ , the expression has different parity:

(a) Come up with an expression that always has even parity. Some examples are: $100p$ and $48w - 2$ . Try to find more.

(b) Come up with expressions that always have odd parity.

(d) The expression $6k + 2$ evaluates to $8, 14, 20, \dots$ (for $k = 1, 2, 3, \dots$ ) — many even numbers are missing.

(e) Are there expressions using which we can list all the even numbers? Hint: All even numbers have a factor $2$ .

(f) Are there expressions using which we can list all odd numbers?

Q19

Context: We saw earlier how to express the $n^{\text{th}}$ term of the sequence of multiples of $4$ , where $n$ is the letter-number that denotes a position in the sequence (e.g., first, twenty third, hundred and seventeenth, etc.).

(1) What would be the $n^{\text{th}}$ term for multiples of $2$ ? Or, what is the $n^{\text{th}}$ even number?

Let us consider odd numbers.

(2) What is the $100^{\text{th}}$ odd number?

To answer this question, consider the following question:

(3) What is the $100^{\text{th}}$ even number?

(4) Write a formula to find the $n^{\text{th}}$ odd number.

Q21

Context: Observe this $3 \times 3$ grid. It is filled following a simple rule — use numbers from $1$ to $9$ without repeating any of them. There are circled numbers outside the grid. The numbers in the yellow circles are the sums of the corresponding rows and columns.

Q. Fill the grids below based on the rule mentioned above:

Q22

Make a couple of questions like this on your own and challenge your peers.

Q27

Can 1 occur in a corner position? For example, can it be placed as follows?

Q. If yes, then there should exist three ways of adding 1 with two other numbers to give 15. We have $1 + 5 + 9 = 1 + 6 + 8 = 15$ . Is any other combination possible?

Q. Similarly, can 9 can be placed in a corner position?

Q29

Can you find the other possible positions for 1 and 9?

Now, we have one full row or column of the magic square! Try completing it!

[Hint: First fill the row or columns containing 1 and 9]

Q30

Choose any magic square that you have made so far using consecutive numbers. If $m$ is the letter-number of the number in the centre, express how other numbers are related to $m$ , how much more or less than $m$ .

[Hint: Remember, how we described a $2 \times 2$ grid of a calendar month in the Algebraic Expressions chapter].

Q31

Context: Choose any magic square that you have made so far using consecutive numbers. If $m$ is the letter-number of the number in the centre, express how other numbers are related to $m$ , how much more or less than $m$ .

[Hint: Remember, how we described a $2 \times 2$ grid of a calendar month in the Algebraic Expressions chapter].

Q. Once the generalised form is obtained, share your observations with the class.

Q32

Chautīsā means 34. Why do you think they called it the Chautīsā Yantra? Every row, column and diagonal in this magic square adds up to 34. Can you find other patterns of four numbers in the square that add up to 34?

Q33

How many rhythms are there with 8 beats consisting of short syllables (1 beat) and long syllables (2 beats)? That is, in how many ways can one fill 8 beats with short and long syllables, where a short syllable takes one beat of time and a long syllable takes two beats of time?

Q34

Context: A short syllable takes one beat of time and a long syllable takes two beats of time. Some possibilities to fill 8 beats are:

long long long long
short short short short short short short short
short long long short long
long long short short long

Q. Can you find others?

Q35

Context: We can write the number $8$ as a sum of $1$ s and $2$ s in several ways, for example:

$8 = 2 + 2 + 2 + 2$
$8 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1$
$8 = 1 + 2 + 2 + 1 + 2$
$8 = 2 + 2 + 1 + 1 + 2$

Q. Do you see other ways?

Q36

Try writing the number 5 as a sum of 1s and 2s in all possible ways in your notebook! How many ways did you find? (You should find 8 different ways!) Can you figure out the answer without listing down all the possibilities? Can you try it for $n = 8$ ?

Q37

Context: Thus, there are 8 rhythms having 5 beats! The reason this method works is that every 5-beat rhythm must begin with either a '1+' or a '2+'. If it begins with a '1+', then the remaining numbers must give a 4-beat rhythm, and we can write all those down. If it begins with a 2+, then the remaining number must give a 3-beat rhythm, and we can write all those down. Therefore, the number of 5-beat rhythms is the number of 4-beat rhythms, plus the number of 3-beat rhythms. How many 6-beat rhythms are there? By the same reasoning, it will be the number of 5-beat rhythms plus the number of 4-beat rhythms, i.e., $8 + 5 = 13$ . Thus, there are 13 rhythms having 6 beats.

Q. Use the systematic method to write down all 6-beat rhythms, i.e., write 6 as the sum of 1's and 2's in all possible ways. Did you get 13 ways?

Q38

Write the next 3 numbers in the sequence: $1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \underline{\quad}, \underline{\quad}, \underline{\quad}, \dots$

If you have to write one more number in the sequence above, can you tell whether it will be an odd number or an even number (without adding the two previous numbers)?

Q39

Context: $1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \dots$

What is the parity of each number in the sequence? Do you notice any pattern in the sequence of parities?

Q40

Context: Let us look at one more example. Here $\text{K2}$ means that the number is a 2-digit number having the digit '2' in the units place and 'K' in the tens place. $\text{K2}$ is added to itself to give a 3-digit sum $\text{HMM}$ :

\begin{array}{r} \text{K2} \\ +\ \text{K2} \\ \hline \text{HMM} \end{array}

Q. What digit should the letter $\text{M}$ correspond to? Both the tens place and the units place of the sum have the same digit. What about $\text{H}$ ? Can it be 2? Can it be 3?

Q41

Context: These types of questions can be interesting and fun to solve! Here are some more questions like this for you to try out. Find out what each letter stands for. Share how you thought about each question with your classmates; you may find some new approaches.

Q. Find out what each letter stands for:

(i) $\begin{array}{r} \text{YY} \\ +\quad \text{Z} \\ \hline \text{ZOO} \end{array}$

(ii) $\begin{array}{r} \text{B5} \\ +\quad \text{3D} \\ \hline \text{ED5} \end{array}$

(iii) $\begin{array}{r} \text{KP} \\ +\quad \text{KP} \\ \hline \text{PRR} \end{array}$

(iv) $\begin{array}{r} \text{C1} \\ +\quad \text{C} \\ \hline \text{1FF} \end{array}$