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For all x E Z, 4lx implies 4lx²

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Final answer:

If a number x is divisible by 4, then x² is also divisible by 4 because squaring x preserves its divisibility properties. This is confirmed by the rules of exponentiation.

Step-by-step explanation:

The question asks to verify if '4lx' implies that '4lx²' for all x belonging to the set of integers (Z). To test this implication, we should recall that if a number is divisible by 4, then it is also divisible by 4 when squared because the square of any integer is that integer multiplied by itself, thus maintaining the divisibility by 4.

This is because the exponentiation of a number is essentially multiplication performed repeatedly. If 4 divides x (4|x), it means x = 4k for some integer k; squaring both sides results in x² = (4k)², which is 16k² (since (2k)² = 4² · k²). This clearly shows that 16k² is divisible by 4, thus proving that 4|x². It follows the rules of exponentiation, where n = n × nx-1, so in our case, 16k² = 4 × (4k²).

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