What can you work out about two numbers from knowing the result of adding them both correctly and incorrectly?

(a) The sum of two positive integers is 111111. A calculator is being used to add the two integers, but, by mistake, an extra 0 is entered at the end of the first integer (and all the other digits are entered correctly). The calculator shows a sum of 441672.

What are the two integers?

(b) The sum of two positive integers is 111111. A calculator is being used to add the two integers, but, by mistake, some extra 0’s are entered at the end of the first integer (and all the other digits are entered correctly). The calculator shows a sum of 1495725.

What are all the possibilities for the two integers?

(c) The sum of two positive integers is 111111. One of the digits of the first integer is zero.

A calculator is being used to add the two integers, but, by mistake, the zero digit in the first integer is entered as 00 instead of 0 (and all the other digits are entered correctly). The calculator shows a sum of 219111.

What would the calculator show for the sum if, by mistake, the zero digit of the first integer is omitted (and all the other digits are entered correctly)?