[updated to correct a typo in b, and clarify d] [and d, again]

So I gave a three digit number to five of my friends and they all told me two facts about it. Unhappily each person gave me one correct fact and one incorrect fact. What was the number?

a1: It is the difference between two squares
a2: It is not the sum of two cubes

b1: It is an even number
b2: It does not have exactly two prime factors (it is not semiprime; ie, not of the form a*b where a and b are both prime)

c1: It is the sum of two squares
c2: It is both a square and a cube

d1: It is a product of three primes, not necessarily distinct
d2: It is prime

e1: It is the sum of a square and a cube
e2: It is the sum of three squares

Source: I think Bertie Taylor from compuserve’s old SCIMAT forum. I don’t know where Bertie took his puzzles from. I posted it on this blog a few years ago, under the title Fallible Friends.

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1. December 1, 2009 pm31 6:27 pm 6:27 pm

b2: do you mean “it does not have exactly two prime factors” or “it is semiprime”? They’re opposites, no?

• December 1, 2009 pm31 6:33 pm 6:33 pm

Actually, no, I’m more confused than that even.

24 has exactly two prime factors: 2, and 3. But it is not semiprime.

If it is semiprime, it has exactly two prime factors.

If it has exactly two prime factors, it may be semiprime but need not be.

Which do you mean? (And, for d1, do you mean “it is pqr, with p,q,r prime”, or what you wrote? Or is it deliberately ambiguous?)