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A group of 25 friends were discussing a large positive integer. "It can be divided by 1," said the first friend. "It can be divided by 2," said the second friend. "And by 3," said the third friend. "And by 4," added the fourth friend. This continued until everyone had made such a comment. If exactly two friends were incorrect, and those two friends said consecutive numbers, what was the least possible integer they were discussing?
i got this problem somewhere. wanted to share it..
Comments (6)
the LCM of the first 25 positive integers is
2^4 * 3^2 * 5^2 * 7 * 11 * 13 * 17 * 19 * 23...
first of all we have to remove two consecutive positive integers.
we will start from the terms from the back.
24*25. but then we already have 5 and 15 and than will mean that we can't take out 25.
since the number is divisible by 5 and 15, it is also divisible by 25.
similalry if 24*23 isn't possible too since we have 8 and 3.
since the number is divisible by 8 and 3 it has to be divisible by 24...
so continue like this. it will be easier to look for primes and the numbers next to them.
we reach 16 and 17.
It is the only number <25 that has four factors of two.
Thus if we say that it isn't divisible by 16, we are saying that it doesn't have four factors of 2. But no other numbers do either, and thus it is legal.
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Is the number 17*13*11*7*25*9*16 = 61261200 ?
I assumed that the ones who said 19, 23 divide the number were wrong...