SOLUTION: if a is an integer, which of the following is always an odd integer? (a)4a+2 (b)10a (c)8a+1 (d).83 please help!!!!!!

Algebra ->  Divisibility and Prime Numbers -> SOLUTION: if a is an integer, which of the following is always an odd integer? (a)4a+2 (b)10a (c)8a+1 (d).83 please help!!!!!!      Log On


   

Question 214411This question is from textbook algebra 1 an integrated approach
: if a is an integer, which of the following is always an odd integer?

(a)4a+2 (b)10a (c)8a+1 (d).83
please help!!!!!! This question is from textbook algebra 1 an integrated approach

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Recall that if 'k' is an integer, then 2k%2B1 is ALWAYS odd. Why? The number 2k is ALWAYS even (because all even integers are a multiple of 2). If you add 1 to an even integer, you will get an odd integer. So this shows us that 2k%2B1 is ALWAYS odd where 'k' is an integer.


So the goal is to try to transform each choice into either the form 2k (even form) or 2k%2B1 (odd form).


a)

4a%2B2 Start with the given expression


2%28a%2B1%29 Factor out the GCF 2


2k Replace 'a+1' with 'k' (which is an integer)


So 4a%2B2 is ALWAYS even (try plugging in some integers for 'a' to test this).

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b)

We can rewrite 10a as 2%2A5a which can come to 2k (where k=5a. So 10a is ALWAYS even.

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c)

We can rewrite 8a%2B1 as 2%2A4a%2B1 and then let k=4a. Now replace '4a' with 'k' to get 2k%2B1. So 8a%2B1=2k%2B1 (where k=4a)

This shows us that 8a%2B1 is ALWAYS odd.

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d)

The value 0.83 is neither even nor odd. Why is this? A number can only be even or odd when it is a whole number or integer. Since 0.83 is NOT a whole number, it is neither even nor odd.



So to recap, the only odd integer of the group is 8a%2B1