1/21/2024 0 Comments Odd numbers![]() The differences between odd and even numbers are tabulated below: Now we will learn the differences between odd and even numbers. We have learnt what are even and odd numbers. The properties of odd and even numbers are tabulated in the table below: These trends result in the properties of the even number and odd numbers. While performing the operations with odd and even numbers we can observe some trend in the result. However, we may list the initial odd numbers, which contain both positive odd numbers like 1, 3, 5, 7, 9, and so on, going to infinity, and negative odd numbers like -1, -3, -5, -7, -9, and so on, reaching to negative infinity. There are an infinite number of odd numbers, hence it is impossible to list them all. Odd numbers can also be negative, as in -81, -35, -55, and so on. ![]() It can be written in form of 2n+1, where n can be any integer number.įor example : 81, 35, 55, 7, and so on are all odd numbers. The approach to identifying odd numbers is that the odd number’s last digit is always 1, 3, 5, 7, or 9. If we are given a number and divide it by two and will result in the remainder is one, then it is an odd number. In mathematics, odd numbers are numbers which when divided by two result a remainder as one. Software Engineering Interview Questions.Top 10 System Design Interview Questions and Answers.Top 20 Puzzles Commonly Asked During SDE Interviews.Commonly Asked Data Structure Interview Questions.Top 10 algorithms in Interview Questions.Top 20 Dynamic Programming Interview Questions.Top 20 Hashing Technique based Interview Questions.Top 50 Dynamic Programming (DP) Problems.Top 20 Greedy Algorithms Interview Questions.Top 100 DSA Interview Questions Topic-wise.This is the case because the reasoning depends only on whether the numbers considered are even or odd. The reasoning in part (b) applies no matter how far the multiplication table is extended. For the same reason, adjacent rows cannot contain odd numbers. Adjacent columns cannot contain odd numbers because one of the columns is an even column (which contains only even numbers) and one is an odd column. We also know that every other column/row is even numbered because the even numbers are found counting by $2$'s which skips one whole number each time. We know from above that even numbered rows and even numbered columns contain only even numbers. So even numbered rows and columns contain only even numbers. Since $4$ can be divided evenly into two pair, all of these numbers can be divided evenly into pairs and so they are all even. Using the commutative property of multiplication, this is the same as $1 \times 4, 2 \times 4, \ldots, 6 \times 4$. If we look, for example, at the second column, this contains the numbers $1 \times 2, 2 \times 2, \ldots, 6 \times 2$: these are all even numbers because we can visualize these as one pair, two pair, $\ldots$, six pairs. For a product such as $6 \times 5$, we could first use the commutative property of multiplication to give $6 \times 5 = 5 \times 6$ and then use the same reasoning we used to see why $4 \times 6$ is even. Similar reasoning will work in other cases. For a larger number such as $24 = 4 \times 6$, a good way to visualize why this is an even number would be to view the number $6$ as made up of $3$ pairs: then multiplying by $4$ means that we take $4$ groups of $6$:Īnd so $4$ groups of $3$ pairs. We can view this, using the distributive property, asīelow the even numbers in the table have been shaded blue:įor smaller numbers such as $2,4,6,8$ students will know from experience that these numbers are even. Students are most likely to benefit if they can work with a partner or in small groups on this task.Īlthough not requested in the problem, teachers may wish to investigate further why a product of an odd number by another odd number appears to always be odd. ![]() In general, the students will probably need a lot of help understanding what the task is asking. The teacher should point out that the first row and first column are not part of the table itself but show what numbers are being multiplied. The multiplication table itself is a little hard to parse because it has so much information in it. Possibility of reaching the number counting by $2$'s or expressing the number as a whole number of pairs. The even numbers in the tableĪre examined in depth using a grade appropriate notion of even, namely the This problem examines the ''checkerboard'' pattern of even and odd numbers The goal is to look for structure and identify patterns and then try to find the mathematical explanation for this.
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