If you add n whole numbers (even or odd) when is the sum of the numbers odd? Translate into adding O's and l's (mod 2). When do they add to 1?

Algebra and Trigonometry (6th Edition)
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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If you add n whole numbers (even or odd) when is the sum of the numbers odd? Translate into adding O's and l's (mod 2). When do they add to 1?

Expert Solution
Step 1

A number is said to be an odd number if it is not divisible by 2 and is of the form 2n±1  where n.

A number is said to be an even number if it is divisible by 2 and is of the form 2n where n.

i.e.,

2n(mod 2)=0   n.and 2n±1(mod 2)=1  n.

Step 2

If we add n whole numbers then  (1) If n is odd and all the numbers are odd then their sum is odd.example :- sum of 3 odd numbers 3+5+7=15 is an odd number.(2) If n is even and all the numbers are odd then their sum is even.example :- sum of 4 odd numbers 3+5+7+3=18 is an even number.(3) If all the numbers are even then their sum is always an even number.example :- sum of 3 even numbers 2+6+10=18 is an even number.and sum of 4 even numbers2+6+10+20=38 is an even number.(4) if some numbers are odd and some numbers are even then their sum is odd if the counting of the odd numbers is odd.example :- sum of some odd numbers and some even numbers 3+5+7+2=17 is an odd number.and 3+5+2=10 is an even number.i.e., the sum of n whole numbers is odd iff the counting of odd numbers is odd in the summation operation.

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