Question 70
Sergio asked Ivan to think of a positive integer, then add its digits and square the result. Without revealing the initial number, Ivan said his final result was x. Show Sergio how to reach these conclusions:
a) If Ivan's number had at most 3 digits, then x would be less than 730.
b) If it had 4 digits, x would be smaller than Ivan's number.
c) If it had 5 or more digits, x would be smaller than Ivan's number.
Sergio then took x, added its digits, and squared the result. When Sergio announced his result, Ivan said in surprise that it was his original number.
d) Determine all possible numbers Ivan could have chosen.
a) If Ivan's number had at most 3 digits, then x would be less than 730.
b) If it had 4 digits, x would be smaller than Ivan's number.
c) If it had 5 or more digits, x would be smaller than Ivan's number.
Sergio then took x, added its digits, and squared the result. When Sergio announced his result, Ivan said in surprise that it was his original number.
d) Determine all possible numbers Ivan could have chosen.