187 – Quiz 1

Here is quiz 1. 

Problem 1 is False. This is because the number 1 is neither prime nor composite. 

Problem 2 is False. This is because we can have x=-y\ne0, in which case x^2=y^2 but x\ne y. For example, consider x=1, y=-1.

For problem 3, start by writing the number n=1\times 2\times \dots\times 9 as a product of primes: n=2^7\times 3^4\times 5\times 7. Plainly, any positive divisor of n must have the form 2^a\times 3^b\times 5^c\times 7^d where a=0,1,\dots, or 7; similarly, b=0,1,\dots, or 4; c=0 or 1, and d=0 or 1. There are 8 possibilities for a, 5 for b, 2 for c, and 2 for d. This gives us a total of 8\times 5\times 2\times 2=160 possible positive divisors.

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