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Professor and Students (Posted on 2003-04-28) Difficulty: 4 of 5
A Professor asked four students how long each of them had been studying.

One of the students replied: “We have all been studying a whole number of years, the sum of our years of studying is equal to the number of years you have been teaching and the product of our years of studying is 180”.

“I’m sorry”, replied the Professor after some thought, “but that doesn’t give me enough information”.

“Yes, you’re right”, agrees another of the students. “But if we told you that one of us were into double figures in our years of study, then you could surely answer your question”.

How long had each of the four been studying ?

See The Solution Submitted by Ravi Raja    
Rating: 3.0000 (4 votes)

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Solution solution | Comment 1 of 10
There are 25 sets of factors of 180, and they could add up as follows:

3 + 3 + 4 + 5 = 15
2 + 3 + 5 + 6 = 16
1 + 5 + 6 + 6 = 18
2 + 2 + 5 + 9 = 18
2 + 3 + 3 +10 = 18
1 + 4 + 5 + 9 = 19
1 + 3 + 6 +10 = 20
1 + 3 + 5 +12 = 21
1 + 2 + 9 +10 = 22
2 + 2 + 3 +15 = 22
1 + 3 + 4 +15 = 23
1 + 2 + 6 +15 = 24
1 + 2 + 5 +18 = 26
1 + 3 + 3 +20 = 27
1 + 1 +12 +15 = 29
1 + 1 +10 +18 = 30
1 + 1 + 9 +20 = 31
1 + 2 + 3 +30 = 36
1 + 1 + 6 +30 = 38
1 + 1 + 5 +36 = 43
1 + 2 + 2 +45 = 50
1 + 1 + 4 +45 = 51
1 + 1 + 3 +60 = 65
1 + 1 + 2 +90 = 94
1 + 1 + 1+180 =183

Only two of the totals, 22 and 18, have more than one set of four factors that add up to them, and so would be ambiguous if the professor's teaching years were equal to one of them. But those adding to 22 both have terms with two digits.

Those adding to 18 have only one with two digits: 2, 3, 3 and 10. So this is the number of years of study of the four students.
  Posted by Charlie on 2003-04-28 03:53:56
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