Replace the interrogation marks in the following statements by appropriate digits (0 to 9) to make these statements valid:

D1: 34?561? is exactly divisible by 45
D2: 68??37 is exactly divisible by 99
D3: 9???057 is equal to 417 times ?1???
D3: 70??34? is exactly divisible by 792
D4: 4?18? is exactly divisible by 101
D5: 6?80?8??51 is exactly divisible by 73*137

73*137=10001 so set up a long multiplication as for D4.

    abcdef
x    10001 
    abcdef
abcdef    
6?80?8??51
Clearly a=6, e=5, f=1 and three of the question marks are b, c, d.

    6bcd51
x    10001 
    6bcd51
6bcd51    
6b80?8cd51

b+1=8 so b=7, 6+5 gives the last ? which is a 1, 1+d ends in 0 so d=9, 1+c=8 so c=7.

    677951
x    10001 
    677951
677951    
6780187951

Posted by Jer on 2011-06-07 10:16:13