Question

a) Using the digits 0 to 7 exactly once each, form a collection of positive integers whose sum is 100.

b) show that in any solution to (a), the sum of the digits in the units places is 20.
c） find, with reasons, the smallest possible value of the largest integer in a solution to (a).

Answer 1

By working with the given set we get:

a) The numbers can be: 23, 16, 7, 54, 0.

b) Because the units can be commuted, their sum is always equal to 20.

c) The smallest is 0 and the largest is 57.

Whose numbers add up to 100?

This can be done by trial and error.

For example, only using the digits {0, 1, 2, 3, 4, 5, 6, 7}

We can make, for example:

23 + 17 + 6 + 54 + 0 = 100

Where the numbers used are:

23

17

6

54

0.

b) If we add the units digits of these 5 numbers, we get:

3 + 7 + 6 + 4 + 0 = 20

Now, if we commute the units digits between the different numbers, to get for example:

27

13

6

54

0

The sum is still equal to 100, and the units digits are the same ones, so we still have that the sum of the units digits is equal to 20.

c) The smallest integer that we use is 0.

For the largest we just commute the tens and units digits so we make the larger integer possible, which is:

57.

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