Eight is Known

Strategies and Hints

The total solution is:



Where two digits are brought down in one step, it is clear that a zero occurs in the quotient. Therefore, we know that the answer must be x080x. Let the divisor be denoted by D. We next note that 8D is a three digit number by looking at the second step of the division algorithm. However, the other multiples of D are four digit numbers. Thus, the quotient must be 90809.

Since 8 · D is a three digit number, it must be less than 1000. We can write
8 · D < 1000, or D < 125.

After determining the first 9 and 0 in the quotient, the different that remains is a six digit number. A six digit number is greater than 99,999. Thus, the rest of the quotient (809) times the divisor (D) must be greater than 99,999. Thus, 809 · D > 99,999 which gives D > 123. If D < 125 and D > 123, and we know D is an integer, the D must be 124. The product of 124 and 80908 is 11,260,316 and the rest of the solution is easily determined.