Question

Statement 3. If m is an even integer, then 3m + 5 is an odd integer.

a. Play with the statement - i.e., Look at/test a few examples. (See if there are any counter-examples.)

b. Write a proof of this statement. (Hint: 5 = 4 + 1 = 2(2) + 1)

Remark 1. Let's recap some important properties about odd and even that we have seen (in notes and this activity): i. If a and b are even, then ab is even.

ii. If a and b are even, then a + b is even.

iii. If a and b are odd, then ab is odd.

iv. If a and b are odd, then a + b is even.

v. If a is even and b is odd, then ab is even.

vi. If a is even and b is odd, then a + b is odd. With this, we can present an alternative proof of the previous statement (here we can make our lives easier by using known results/facts that we have already shown!)

Proof of Statement 3. Let m be an even integer. Since m is even and 3 is odd we have that 3m is even. Moreover, as 5 is odd, it follows that 3m + 5 is odd.

Answer 1

(a). For m=2 , 3m+1 = 7

For m=4 , 3m+ 1 = 13

For m=6 , 3m+1 = 19

All are odd . So possibly there are no counter example . .

(b). Proof : Since m is even integer so there exist an integer k such that m=2k.  

Now , 3m + 5

, where l = 3k +2 is an integer .

So 3m + 5 can be written as of the form . Hence is an odd integer .

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If you have doubt at any step please comment .