Question

4. If p and q are two consecutive odd positive integers such that p > q then prove that the two
p+q P-9
numbers
and are even and odd integers respectively.
2
2
5. (a) Show that any positive odd integer is of the form 49 + 1 or 49 +3, where q is some integer.​

Answer 1

Step-by-step explanation:

As we know, even numbers are always divisible by 2. So if any number is divisible by 2, it is an even number.

Even & odd number come alternatively om number line. So, if 2a is even, 2a + 1 or 2a + any odd number, must be odd.

Now, let q = 2a + 1 & p(>q) = 2b + 1.

So,

p + q = 2a + 1 + 2b + 1 = 2(a + b) + 2

p + q = 2(a + b + 1)

As p + q is divisible by 2, it is an even number.

p - q = 2b + 1 - (2a - 1) = 2b + 1 - 2a - 1

p - q = 2(b - a)

p - q = number divisible by 2, it is also an even number. Check your quetion.

(5): As mentioned above ' if any number is divisible by 2, it is an even number.

Even & odd number come alternatively om number line. So, if 2a is even, 2a + 1 or 2a + any odd number, must be odd',

Similarly, 4a is even, 4a + 1 or 4a + any odd number, must be odd.

Thus, 4q + 1 and 4q + 3, both are odd numbers.