Parity: Show that the sum contains at least one even number.

In summary, parity refers to the property of a whole number being either even or odd. To prove that the sum contains at least one even number, you can use the fact that when you add an even number and an odd number, the result is always odd. An example of a sum that does not contain at least one even number is 3+5+7=15. The importance of proving this statement is to help identify patterns and properties of numbers, and solve mathematical problems. Another way to prove the statement is by using the pigeonhole principle, which states that if you have n+1 objects distributed into n boxes, at least one box must contain more than one object.
  • #1
trashrama
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Hello, guys!

I have a question that need help!

A number with 17 digits is chosen and the order of its digits is inverted, forming a new number, These two numbers are then added up. Show that the sum contains at least one even number.
 
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  • #2
Since the number chosen has 17 digits, its units digit must be odd. When the order of the digits is inverted, the new number's units digit is still odd. Therefore, when the two numbers are added together, the sum must have at least one even number.
 

1. What is parity?

Parity refers to the property of a number being either even or odd.

2. How do you show that a sum contains at least one even number?

To show that a sum contains at least one even number, we can use the fact that the sum of two odd numbers is always even. Therefore, if a sum contains only odd numbers, the result would also be odd. Hence, if the sum is even, it must contain at least one even number.

3. Can you give an example of a sum that contains at least one even number?

One example of a sum that contains at least one even number is 3 + 5 + 7 = 15. This sum contains two odd numbers (3 and 5) and one even number (7). The result, 15, is also an odd number.

4. Is it possible for a sum to contain no even numbers?

No, it is not possible for a sum to contain no even numbers. As mentioned earlier, the sum of two odd numbers is always even. Therefore, any sum that contains only odd numbers would have an even result.

5. Why is it important to show that a sum contains at least one even number?

Showing that a sum contains at least one even number is important because it helps us determine the parity of the result. This information can be useful in various mathematical and scientific calculations and proofs.

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