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Answer :
Final answer:
The square of a number that when divided by 36 leaves a remainder of 3 is of the form (36k + 3)2. When this square is divided by 9, it leaves a remainder of 0 because all the terms in the expansion are multiples of 9.
Explanation:
If a number when divided by 36 leaves a remainder of 3, we can express this number as 36k + 3 where k is an integer. When the square of this number is calculated, we get (36k + 3)², which equals 1296k² + 216k + 9. We are interested in the remainder when this expression is divided by 9.
Let's break down the square of the number: 1296k² is divisible by 9 since 1296 is a multiple of 9, and 216k is divisible by 9 for the same reason. However, when we look at the constant term, which is 9, it is also a multiple of 9. This means that when (36k + 3)² is divided by 9, the remainder comes from the term 9, which is a multiple of 9, resulting in a remainder of 0.
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