Challenge Level

We are using $ABC$ to represent a 3 digit number with $A$ hundreds, $B$ tens and $C$ units. Similarly, $AB$ represents a 2 digit number with $A$ tens and $B$ units.

**Problem 1:**

A positive 2 digit integer $AB$ is four times the sum of its digits.

Prove that there are exactly four such numbers $AB$.

Prove that there are exactly four such numbers $AB$.

**Problem 2:**

Show that there is only one set of values $A, B$ and $C$ that satisfies $$ABC + AB + C = 300$$

**Problem 3:**

Choose a two-digit number with two **different** digits (e.g. $47$) and form its reversal (i.e. 74). Now, subtract the sum of the digits from each of these numbers, and then add the two results. Show that you always obtain a multiple of $9$.

**Problem 4:**

Choose three different digits and form the six two-digit numbers that use two of the three digits. Add these six possibilities and divide this total by the sum of the three digits. Show that you always obtain $22$.

**Problem 5:**

Choose three different digits $A, B$ and $C$, with $A > B > C$ (e.g. $8, 6$ and $3$).

Work out the differences between the two-digit numbers you can make and their reverses (e.g. $86-68; 63-36; 83-38$), then add these three results.

Show that you always obtain a multiple of $18$.

Work out the differences between the two-digit numbers you can make and their reverses (e.g. $86-68; 63-36; 83-38$), then add these three results.

Show that you always obtain a multiple of $18$.

**Problem 6:**

For any three digits $A, B$ and $C$, show that there is only set of values that satisfies $$ABC=AB+BC+CA.$$

**Problem 7:**

Find examples that fit the rule: $$AB+CD=DC+BA,$$ without any of the four digits being the same (e.g. $97+24=42+79$).

What general rule must apply? Why?

*With thanks to Don Steward, whose ideas formed the basis of this problem.*

What general rule must apply? Why?