13.3 A division problem 405
Another division problem: Problem E1 of theMonthly
x 7 xxx
xxx) xxxxxxxx
xxxx
xxx
xxx
xxxx
xxx
xxxx
xxxx
Clearly, the last second digit of the quotient is 0.
Let the divisor be the 3-digit numberd.
Consider the 3-digit number in the seventh line, which is a multiple of
d. Its difference from the 4-digit number in the sixth line is a 2-digit
number. This must be 9 xx.
This cannot be the same as the 3-digit number in the fifth line, since the
difference between the 3-digit numbers in the fourth and fifth lines is a
3-digit number.
Therefore, in the quotient, the digit after 7 is a larger one, which must be
smaller than the first and the last digits, since these give 4-digit multiples
ofd.
It follows that the quotient is 97809.
Since 8 dis a 3-digit number 9 xx, the 4-digit number in the third and
bottom lines is 9 d=10xxor 11 xx.
From this 8 dmust be 99 x, and therefore992 = 8× 124.
97809
124)12128316
1116
968
868
1003
992
1116
1116