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# Two positive integers and are chosen

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Two positive integers m and n are chosen such that m is the largest positive integer less than 100 with only two positive divisors and n is the smallest integer with exactly three positive divisors. What is m + n?

Nov 20, 2020

#1
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Two positive integers m and n are chosen such that m is the largest positive integer less than 100 with only two positive divisors and n is the smallest integer with exactly three positive divisors. What is m + n?

If the number is to have only two divisors,

then at least one of the divisors has to be prime.

So how about taking the largest prime less than 50,

and doubling it, so that would be 47 x 2, thus m = 94

If 1 is qualified to be considered a divisor, then m = 97 but I don't like that.

The other number, smallest with three divisors, I just used the

three smallest factors, so, that would be 2 x 3 x 5, thus n = 30

Final answer, m + n = 124

If you wanted to qualify 1 as a divisor, you can find n the same way.

Nov 20, 2020
#2
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The largest positive integer less than 100 with only 2 positive divisors = 97 [ 1 and 97]

The smallest integer with exactly 3 positive divisors = 4 [1,  2  and 4]

[97  +  4] = 101

Nov 20, 2020