How do you prove that the product of an odd and even number is even?
Theorem: The product of an even integer and an odd integer is even. Proof: Let a and b be integers. Assume a is even and b is odd, so there exists an integer p so that a=2p and there exists an integer q so that b=2q+1. If a⋅b is even then by definition of even there exists an integer r such that a⋅b=2r.
Why is the product of 2 times any number is an even number?
EVEN NUMBERS can be looked at as any number (call it “n”), multiplied by 2. Therefore, all even numbers can be described as 2n. Therefore, any even number plus any other even number will always equal an even number (as the answer you get will always be some number multiplied by two).
What can you say about the product if one is odd and one is even?
the product of an odd number and an even number is always odd.
How do you prove that the product of two rational numbers are rational?
“The product of two rational numbers is rational.” So, multiplying two rationals is the same as multiplying two such fractions, which will result in another fraction of this same form since integers are closed under multiplication. Thus, multiplying two rational numbers produces another rational number.
Why do 2 odds make an even?
simple answer is that between every odd number there is one digit difference. so when you add one and one its become two that is even. That’s why the addition of odd numbers become even.
What kind of number is the sum of two odd numbers?
even
The sum of two odd numbers is always even. The product of two or more odd numbers is always odd. The sum of an even number of odd numbers is even, while the sum of an odd number of odd numbers is odd.
Is the sum of two even numbers always even?
Yes, the sum of two even numbers is always even. So, x and y are mutiples of 2. x=2a and y=2b ; where a and b are any two integers. This means x+y is also multiple of 2. So x+y is an even number. Bonus – The sum of two odd numbers is also always even.
What is the difference between an even and odd integer?
– Odd integer = 2n + 1, where n is any integer. – Even integer = 2n, where n is any integer. You’ve gotta have firm definitions before you start. An example: Pick an odd and even integer, say 3 and 12. n’s obviously different here:
Is (2mn + m + n) an even or odd integer?
Since (2mn + m + n) will be an integer by closure we can conclude that the product ab is an odd integer, which contradicts the assumption. This means at least one of the integers a or b must be even. QED? elementary-number-theorydiscrete-mathematicsproof-verification Share Cite Follow edited Feb 17 ’14 at 6:39 NNOX Apps 1
Is the sum of 2m and 2n even?
Therefore sum is even. Let a = 2a’ and b=2b’. Then a+b=2a’ +2b’ = 2 (a’+b’), and 2 (a’+b’) |2 = 0, qed. Since the definition of an even number is that it has a factor of 2, call the two even numbers 2m and 2n, for some m and n.