1. UPC detective: find the missing number in each of these UPC codes:

Code

missing digit

 

A. ?-78787-03104-3

0

 

B. 0-?0735-00458-1

7

C. 0-7?787-04104-2

8

 

D. 0-41?40-56400-3

5

 

E. 0-875?7-79000-7

4

 

F. 0-4154?-33111-7

0

 

G. 0-51515-?2385-0

5

 

H. 0-41540-5?401-0

6

 

I. 0-30643-54?23-7

3

 

J. 0-21200-665?2-6

2

 

K. 0-71641-8000?-5

4

 

L. 0-41540-3351?-6

4

 

B. 0-?0735-00458-1

3-1-3-1 way:

0x3+N+0x3+7+3x3+5+0x3+0+4x3+5+8x3+1=0
N+7+9+5+12+5+24+1=0
63+N=0
3+N=0
N=0-3=7
The missing digit is 7

7-9-7-9 way:
0x7+Nx9+0x7+7x9+3x7+5x9+0x7+0x9+4x7+5x9+8x7=1
9N+63+21+45+28+45+56=1
9N+3+1+5+8+5+6=1
9N+28=1
9N=1-8=0-7=3
9N=3
(9 times what has the last digit of 3? 9x7=63 so)
N=7

C. 0-7?787-04104-2

The 3-1-3-1 way
0x3+7+Nx3+7+8x3+7+0x3+4+1x3+0+4x3+2=0
7+Nx3+7+24+7+4+3+12+2=0
66+Nx3=0
3N+6 =0
3N =0-6
3N=4
(3 times what has the last digit 4? 3x8=24 so)
N=8

 The 7-9-7-9 way
0x7+7x9+Nx7+7x9+8x7+7x9+0x7+4x9+1x7+0x9+4x7=2
63+7N+63+56+63+36+7+28=2
3+7N+3+6+3+6+7+8=2
7N+36=2
7N+6=2
7N=2-6=0-4
7N=6
(7 times what has the last digit 6? 7x8=56 so)
N=8

 

E. 0-875?7-79000-7

The 3-1-3-1 way:
0x3+8+7x3+5+Nx3+7+7x3+9+0x3+0+0x3+7=0
8+21+5+3N+7+21+9+7=0
3N+78=0
3N+8 =0
3N=0-8=2
(what times 3 has the last digit 2? 4x3=12, so)
N=4		 The 7-9-7-9 way
0x7+8x9+7x7+5x9+Nx7+7x9+7x7+9x9+0x7+0x9+0x7=7
72+49+45+7N+63+49+81=7
2+9+5+7N+3+9+1=7
7N+29=7
7N+9=7
7N=7-9=0-2
7N=8
(7 times what has the last digit 8? 7x4=28, so)
N=4