The product of all the prime numbers between 1 and 100 is equal to $P$. What is the remainder when $P$ is divided by 16?
I have no idea how to solve this, any answers?
[Math] the remainder when the product of the primes between 1 and 100 is divided by 16
elementary-number-theory
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Best Answer
Skip the first prime $2$ and look for the product modulo $8$. The twenty-four odd primes $<100$ are $$3,5,7,11,13,17, 19,23,29,31,37,41, 43,47,53,59,61,67, 71,73,79,83,89,97. $$ Modulo $8$, these are $$3,5,7,3,5,1,3,7,5,7,5,1,3,7,5,3,5,3,7,1,7,3,1,1$$ so their product is $$1^53^75^67^6\equiv 3\pmod 8 $$ (where we might profit from using $x^2\equiv 1\pmod 8$ for odd $x$). Thus $P\equiv 6\pmod{16}$.