We know that: (a) even + even = even (b) odd + odd = even (c) even + odd = odd Similarly, find out the parity for the scenarios below: (d) even - even = (e) odd - odd = (f) even - odd = (g) odd - even =

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Question 5

We know that:

(a) even + even = even

(b) odd + odd = even

Similarly, find out the parity for the scenarios below:

(d) even - even = _________

(e) odd - odd = _________

(f) even - odd = _________

(g) odd - even = _________

Solution

We will find the parity of subtraction results using examples and number definitions.

Step 1 — What are Even and Odd Numbers? Let us first remember what even and odd numbers are. An even number can be divided by 2 with no remainder. We can write an even number as $2 \times k$ . Here, $k$ is any whole number. For example, $2 \times 3 = \textbf{6}$ (even). An odd number leaves a remainder of 1 when divided by 2. We can write an odd number as $2 \times k + 1$ . Here, $k$ is any whole number. For example, $2 \times 3 + 1 = \textbf{7}$ (odd).

Step 2 — Even minus Even Let us take two even numbers. Let the first even number be 6. Let the second even number be 2. We subtract the second number from the first. $6 - 2$ $= 4$ The number 4 is an even number. Let us use our definitions for any even numbers. Let the first even number be $2k$ . Let the second even number be $2m$ . We subtract the second number from the first. $2k - 2m$ $= 2 \times (k - m)$ Since $k$ and $m$ are whole numbers, $(k - m)$ is also a whole number. So, $2 \times (k - m)$ is an even number.

$\boxed{\text{even - even = even}}$

Step 3 — Odd minus Odd Let us take two odd numbers. Let the first odd number be 7. Let the second odd number be 3. We subtract the second number from the first. $7 - 3$ $= 4$ The number 4 is an even number. Let us use our definitions for any odd numbers. Let the first odd number be $2k + 1$ . Let the second odd number be $2m + 1$ . We subtract the second number from the first. $(2k + 1) - (2m + 1)$ $= 2k + 1 - 2m - 1$ $= 2k - 2m$ $= 2 \times (k - m)$ Since $k$ and $m$ are whole numbers, $(k - m)$ is also a whole number. So, $2 \times (k - m)$ is an even number.

$\boxed{\text{odd - odd = even}}$

Step 4 — Even minus Odd Let us take an even number and an odd number. Let the even number be 8. Let the odd number be 3. We subtract the odd number from the even number. $8 - 3$ $= 5$ The number 5 is an odd number. Let us use our definitions for any even and odd numbers. Let the even number be $2k$ . Let the odd number be $2m + 1$ . We subtract the odd number from the even number. $2k - (2m + 1)$ $= 2k - 2m - 1$ $= 2 \times (k - m) - 1$ This number is 1 less than an even number. So, $2 \times (k - m) - 1$ is an odd number.

$\boxed{\text{even - odd = odd}}$

Step 5 — Odd minus Even Let us take an odd number and an even number. Let the odd number be 7. Let the even number be 2. We subtract the even number from the odd number. $7 - 2$ $= 5$ The number 5 is an odd number. Let us use our definitions for any odd and even numbers. Let the odd number be $2k + 1$ . Let the even number be $2m$ . We subtract the even number from the odd number. $(2k + 1) - 2m$ $= 2k - 2m + 1$ $= 2 \times (k - m) + 1$ This number is 1 more than an even number. So, $2 \times (k - m) + 1$ is an odd number.

$\boxed{\text{odd - even = odd}}$

Answer

(d) even - even = even (e) odd - odd = even (f) even - odd = odd (g) odd - even = odd

More questions in FIO

Arrange the stick figure cutouts given at the end of the book or draw a height arrangement such that the sequence reads:

(a) 0, 1, 1, 2, 4, 1, 5

(b) 0, 0, 0, 0, 0, 0, 0

(d) 0, 1, 0, 1, 0, 1, 0

(e) 0, 1, 1, 1, 1, 1, 1

(f) 0, 0, 0, 3, 3, 3, 3

For each of the statements given below, think and identify if it is Always True, Only Sometimes True, or Never True. Share your reasoning.

(a) If a person says '0', then they are the tallest in the group.

(b) If a person is the tallest, then their number is '0'.

(d) If a person is not first or last in line (i.e., if they are standing somewhere in between), then they cannot say '0'.

(e) The person who calls out the largest number is the shortest.

(f) What is the largest number possible in a group of 8 people?

Using your understanding of the pictorial representation of odd and even numbers, find out the parity of the following sums:

(a) Sum of 2 even numbers and 2 odd numbers (e.g., even + even + odd + odd)

(b) Sum of 2 odd numbers and 3 even numbers

(d) Sum of 8 odd numbers

Lakpa has an odd number of ₹1 coins, an odd number of ₹5 coins and an even number of ₹10 coins in his piggy bank. He calculated the total and got ₹205. Did he make a mistake? If he did, explain why. If he didn't, how many coins of each type could he have?

We know that:

(a) even + even = even

(b) odd + odd = even

Similarly, find out the parity for the scenarios below:

(d) even - even = _________

(e) odd - odd = _________

(f) even - odd = _________

(g) odd - even = _________

How many different magic squares can be made using the numbers 1 – 9?

Create a magic square using the numbers 2 – 10. What strategy would you use for this? Compare it with the magic squares made using 1 – 9.

Take a magic square, and

(a) increase each number by 1

(b) double each number

In each case, is the resulting grid also a magic square? How do the magic sums change in each case?

What other operations can be performed on a magic square to yield another magic square?

Q10

Discuss ways of creating a magic square using any set of 9 consecutive numbers (like 2 – 10, 3 – 11, 9 – 17, etc.).

Q11

Using this generalised form, find a magic square if the centre number is 25.

Q12

What is the expression obtained by adding the 3 terms of any row, column or diagonal?

Q13

Write the result obtained by— (a) adding 1 to every term in the generalised form. (b) doubling every term in the generalised form

Q14

Create a magic square whose magic sum is 60.

Q15

Is it possible to get a magic square by filling nine non-consecutive numbers?

Q16

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Q17

Liswini has a large old encyclopaedia. When she opened it, several loose pages fell out of it. She counted 50 sheets in total, each printed on both sides. Can the sum of the page numbers of the loose sheets be 6000? Why or why not?

Q18

Here is a 2 × 3 grid. For each row and column, the parity of the sum is written in the circle; 'e' for even and 'o' for odd. Fill the 6 boxes with 3 odd numbers ('o') and 3 even numbers ('e') to satisfy the parity of the row and column sums.

Q19

Make a 3 × 3 magic square with 0 as the magic sum. All numbers can not be zero. Use negative numbers, as needed.

Q20

Fill in the following blanks with 'odd' or 'even':

(a) Sum of an odd number of even numbers is ______

(b) Sum of an even number of odd numbers is ______

(d) Sum of an odd number of odd numbers is ______

Q21

What is the parity of the sum of the numbers from 1 to 100?

Q22

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Q23

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Q24

What is the parity of the 20th term of the Virahāṅka sequence?

Q25

Identify the statements that are true.

(a) The expression $4m - 1$ always gives odd numbers. (b) All even numbers can be expressed as $6j - 4$ . (c) Both expressions $2p + 1$ and $2q - 1$ describe all odd numbers. (d) The expression $2f + 3$ gives both even and odd numbers.

Q26

Solve this cryptarithm:

\begin{array}{rcc} & \text{U} & \text{T} \\ + & \text{T} & \text{A} \\ \hline \text{T} & \text{A} & \text{T} \end{array}