A class in number theory was to divide itself into groups of equal sizes to stud

Question

A class in number theory was to divide itself into groups of equal sizes to study the Chinese Remainder Theorem. When the class was divided into groups of 3, two students were left out; when into groups of 4, one was left out. When it was divided into groups of five, the students found that if the professor was added to one of the groups, no one was left out. Since the professor had never really understood the Chinese Remainder Theorem when he was in college, the last arrangement worked out nicely. How many students were there in the class?

Explanation / Answer

Let N be number of students.

Hence,

N=2 mod 3

N=1 mod 4

N=4 mod 5

N=3x+2.

Case 1: x=4m+1

N=3(4m+1)+2=12m+5=1(mod 4)

Case 2: x=4m+2

N=3(4m+2)+2=12m+8=0 mod 4

Case 3: x=4m+3

N=3(4m+3)+2=12m+11=3 mod 4

Case 4: x=4m

N=3*4m+2=12m+2=2 mod 4

Hence, x=4m+1, m=0,1,2,3,

Hence, N=3(4m+1)+2=12m+5

Now we consider remainders w.r.t. 5

Case 1. m=5k

N=12*5k+5=0 mod 5

Case 2. m=5k+1

N=12(5k+1)+5=12 mod 5=2 mod 5

Case 3. m=5k+2

N=12(5k+2)+5=12*2 mod 5=4 mod 5

Case 4. m=5k+3

N=12(5k+3)+5=12*3 mod 5=1 mod 5

Case 5. m=5k+4

N=12(5k+4)=12*4 mod 5=3 mod 5

Hence, m=5k+2, k=0,1,2,3

Hence, N=12(5k+2)+5=60k+29

Hence number of students in class could be:

29,29+60,29+2*60,..

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