Question

1. In a positive integer, each digit after the leftmost one is larger than the digit to its left. Its
square, which is a six-digit number, also has the same property. Find the original positive
integer.

Answer 1

Let the original positive integer be of the form ABCDE, where A is the leftmost digit and B, C, D, E are its subsequent digits, with B > A, C > B, D > C, and E > D.

The largest possible value for A is 9, as A cannot be less than any of its subsequent digits.

The smallest possible value for B is A+1, and the largest possible value for B is 9, to maintain the relationship B > A.

Similarly, the smallest possible value for C is B+1, and the largest possible value is 9.

For D and E, the smallest values are C+1 and D+1 respectively, and the largest values are 9.

Therefore, we have the range of values for ABCDE as:

* 91234 <= ABCDE <= 98765

Squaring the largest possible value of ABCDE yields 975308641, which has seven digits instead of the required six digits. So the largest possible value of ABCDE is 98765.

The smallest possible value of ABCDE is 12345, and its square is 152399025, which satisfies the conditions.

Therefore, the original positive integer is 12345.