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use division algorithm to show that the square of any positive integer is of the form of 5m or 5m+1 or 5m+4 |
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Answer» Answer: Solution: Any number can be represented by the form 5m+r , where r can be 0,1,2,3,4 and m ∈ N Let Q be a POSITIVE integer. By Euclid's Division LEMMA, Q = 5m + r Squaring Q, Q² = (5m + r)² = 25m² + 10mr + r² Q² = 5 ( 5m² + 2mr) + r² We can TAKE 5m² + 2mr as a number K. Then Q² becomes, Q² = 5K + r². Since r ∈ { 0, 1, 2, 3, 4} r² ∈ {0,1,4,9,16} We also have condition that any number of the form aq + r , r ≤ a , since r is the remainder when the number is divided by a. THEREFORE r² < 5 Possible values of r² = { 0 , 1, 4 } Therefore any positive integer is of the form, 5m , 5m +1 or 5m+4. PLZ MARK AS BRIANLIEST,FLW ME AND THX FOR THE SUPERB QUESTION |
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