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4. If p and q are two consecutive odd positive integers such that p > q then prove that the twop+q P-9numbersand are even and odd integers respectively.225. (a) Show that any positive odd integer is of the form 49 + 1 or 49 +3, where q is some integer. |
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Answer» <P>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. |
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