## 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.