Question

An eccentric baseball card collector wants to distribute her collection among her descendants. If she divided her cards among her 17 great-great-grandchildren, there would be 3 cards left over. If she divided her cards among her 16 great-grandchildren, there would be 10 cards left over. If she divided her cards among her 11 grandchildren, there would be 4 cards left over. If she divided her cards among her 7 children, there would be no cards left over. What is the smallest amount of cards needed?

Answer 1

1. The problem is a number theory problem, requiring basic knowledge of modular arithmetic.

2. Let A be the amount of cards.

Then in modular arithmetic : 

i)        A= 3 (mod 17)
ii)       A= 10 (mod 16)
iii)      A=4 (mod 11)
iv)      A= 0 (mod 7)

3. A= 3 (mod 17) means that A=17a+3 for some natural number a.
    
    Substitute A=17a+3 in (ii):
     17a+3=10 (mod 16)
             a=7  (mod 16), so a=16m+7 for some natural number m.

Now A=17a+3=17(16m+7)=272m+122

Substitute A=272m+122 in (iii):

                 272m+122 = 4 (mod 11)
                       8m+1=4 (mod 11)
                           8m=3 (mod 11)

multiply by 4:  32m=12 (mod11)
                       -m=1 (mod 11)
                        m=-1=10 (mod 11)     so m=11t+10 for some t.


4. A=272m+122=272(11t+10)+122=2992t+2842

5. Keep in mind that 2992 = 17*16*11

    2842=2800+42, both 2800 and 42 are multiples of 7, so 2842 is a multiple of 7.

6. A=2992t+2842 is a multiple of 7, 2842 is a multiple of 7 so 2992 must be a multiple of 7.

2992t=17*16*11*t is the smallest possible multiple of 7 with t=7.

7. So A=17*16*11*7+2842=23786