Welcome to M-World, where the only numbers that exist are positive integers that leave a remainder of 1 when divided by 4. In other words, the only M-numbers that exist are
{1, 5, 9, 13, 17, 21, . . .}
(Another description is that these are the numbers of the form 4t + 1 for t = 0, 1, 2, . . ..) In the M-World, we cannot add numbers, but we can multiply them, since if a and b both leave a remainder of 1 when divided by 4 then so does their product. (Do you see why this is true?)
We say that m M-divides n if n = mk for some M-number k. And we say that n is anM-prime if its onlyM-divisors are 1 and itself. (Of course, we don’t consider 1 itself to be an M-prime.)
(a) Find the first six M-primes.
(b) Find an M-number n that has two different factorizations as a product of M-primes.
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