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Professor and Students (Posted on 2003-04-28)

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 ?

Submitted by Ravi Raja

Rating: 3.4000 (5 votes)

Solution: (Hide)

Let the years of study of the four students be a,b,c and d respectively.
Given that their product is equal to 180.
And that the sum of their years of study is equal to the number of years the Professor has been teaching.
All positive factors of 180 are 1,2,2,3,3 and 5.
We can form groups of four numbers such that the product of the four numbers in each group is equal to 180. All possible groups (irrespective of the order of the numbers in each group) are given below along with the sums of the numbers in each of the groups:
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
Now, since the Professor was not able to tell the years of study of his four students, even after knowing their sum, implies that the sum has to be either equal to 18 or 22, in which case he had the following possibilities:
1 + 5 + 6 + 6 = 18
2 + 2 + 5 + 9 = 18
2 + 3 + 3 +10 = 18
1 + 2 + 9 +10 = 22
2 + 2 + 3 +15 = 22
But the moment he knows that one of the students is in double years of study, he can at once conclude that the four students have been studying for 2, 3, 3 and 10 years.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
No Subject	Rosepark	2023-09-16 01:49:46
Dissertation writing service	Adrian45	2023-07-26 05:38:13
No Subject	Hugo Hoysted	2021-07-16 07:17:08
Education	hepev92013	2020-10-29 07:27:48
answer	K Sengupta	2007-11-23 03:54:58
re: solution. Factor of primes...lucky? Charlie	john	2005-06-27 19:52:27
re(3): well -- the computer language	Charlie	2003-05-03 04:43:16
re(2): well -- the computer language	DJ	2003-04-30 16:36:30
Two possible solutions?	arbor	2003-04-30 11:51:12
re(2): well -- the computer language	Gamer	2003-04-29 12:16:14
re: well -- the computer language	Charlie	2003-04-28 16:58:40
well	DJ	2003-04-28 10:32:48
re: solution	Charlie	2003-04-28 03:59:00
solution	Charlie	2003-04-28 03:53:56

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