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We have six statements about a number, and we know that exactly 1 is false.

1. I am greater than 50
2. I am a multiple of 7
3. I am a perfect square
4. I am a 3-digit number
5. I am less than 500
6. I am a multiple of 17

Are there any numbers that fit? How many? And if they exist, what are they?

Unrelated, interesting note: Nice factoring techniques for solving problems such as

Find n such that n(n+16) is a perfect square

are presented at the Ultimate Quant Marathon Blog for IIM Cat (whatever that means), a brand new blog. I think it’s called Quantologic for short.

1. August 31, 2008 pm31 8:06 pm 8:06 pm

I’m assuming ‘number’ = positive. First, which exactly-one is false?

#1 can’t be false if #4 is true.

Suppose #2 is false. X is between 100 and 499, a perfect square, and a multiple of 17 but not 7: 17^2 = 289.

Suppose #3 is false. X is a multiple of 17 and 7 between 101 and 499, not a perfect square. 17*7*i=119,238,357,476.

Suppose #4 is false. X is between 50 and 99 and a perfect-square multiple of 17 and 7; no such X.

Suppose #5 is false. X is between 500 and 999 and a perfect-square multiple of 17 and 7, i.e. (17*7*i)^2 for some i. No such X.

Suppose #6 is false. X is between 100 and 499, a perfect-square multiple of 7 but not 17: X=(7i)^2 = 196, 441.

2. September 1, 2008 am30 2:33 am 2:33 am

I had a similar solution, but narrowed down to three cases at the start:

A number which is a perfect square, and also a multiple of both primes 7 and 17 would have to be a multiple of 7^2×17^2=14161. But clearly no such number can satisfy either condition 4 or condition 5.

Thus we know that one of 2, 3, or 6 is false, and that 1, 4, and 5 must be true.
(etc… as in Sonic Charmer’s attack on not #2, not #3, and not #6.)

3. September 2, 2008 pm30 8:47 pm 8:47 pm

7, 17 and “perfect square” are incompatible, since the smallest such is $(7*17)^2=14161$, which is neither three-digit nor less then 500. So one of these three is the falsehood.