Patterns in the digits

A number like $5{,}832$ is not just one thing , it is a string of digits: $5$, $8$, $3$, $2$. When we play with those digits, surprises pop out.

Concept

Every whole number written in our familiar system uses ten digits: $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$. The place of a digit tells us its value. In $5{,}832$:

• $5$ is in the thousands place, worth $5 \times 1000 = 5000$.
• $8$ is in the hundreds place, worth $800$.
• $3$ is in the tens place, worth $30$.
• $2$ is in the ones place, worth $2$.
• So $5{,}832 = 5000 + 800 + 30 + 2$.

This is place value, and it lets digits work as a team.

Now for some games. The digit sum of a number is what you get by adding all its digits. For $5{,}832$ the digit sum is $5 + 8 + 3 + 2 = 18$. If you take the digit sum of $18$ you get $1 + 8 = 9$. Mathematicians call this final one-digit answer the digital root.

Digit sums are not just play. They give a quick test for divisibility by $3$ and $9$:

• A number is divisible by $9$ if its digit sum is divisible by $9$.
• A number is divisible by $3$ if its digit sum is divisible by $3$.

Try $1{,}998$: digit sum $1+9+9+8 = 27$, and $27$ is divisible by $9$. So $1{,}998$ is divisible by $9$ , check: $1998 \div 9 = 222$.

Another playful pattern: reverse and subtract. Take any 3-digit number, like $652$. Reverse the digits to get $256$. Subtract the smaller from the bigger: $652 - 256 = 396$. Reverse and add: $396 + 693 = 1089$. Try it with $341$ or $874$ , you keep landing on $1089$! This is sometimes called Kaprekar's magic for $3$ digits.

For $4$-digit numbers there is Kaprekar's constant, $6174$. Take any $4$-digit number with at least two different digits , say $3215$. Rearrange to make the biggest and smallest possible: $5321$ and $1235$. Subtract: $5321 - 1235 = 4086$. Repeat: rearrange $4086$ to $8640$ and $0468$, subtract: $8640 - 468 = 8172$. Keep going. Eventually, no matter what number you started with, you reach $6174$ and get stuck there.

There are dozens of such digit games. They show that numbers, like atoms, have hidden structure waiting to be poked.

Worked examples

Example 1. Find the digit sum and digital root of $4{,}769$.

• Digit sum $= 4 + 7 + 6 + 9 = 26$.
• Digital root: $2 + 6 = 8$.

Example 2. Is $7{,}425$ divisible by $9$?

• Digit sum $= 7 + 4 + 2 + 5 = 18$. Divisible by $9$? Yes.
• So $7{,}425$ is divisible by $9$. Check: $7425 \div 9 = 825$. ✓

Example 3. Take the number $352$. Apply the "reverse and subtract" step.

• Reverse: $253$.
• Subtract: $352 - 253 = 99$ (or $099$).
• Reverse the result and add: $099 + 990 = 1089$. Magic!

Example 4. Show that any 2-digit number minus its reverse is a multiple of $9$.

• Let the number be $\overline{ab} = 10a + b$. Reverse $= \overline{ba} = 10b + a$.
• Difference $= (10a + b) - (10b + a) = 9a - 9b = 9(a - b)$.
• So it is always a multiple of $9$.

Try it yourself

1. Find the digit sum and digital root of $8{,}461$.
2. Is $3{,}426$ divisible by $3$? By $9$?
3. Try the "reverse and subtract then reverse and add" trick on $874$. Do you reach $1089$?
4. Find the largest 4-digit number whose digit sum is $9$.
5. Find the smallest 4-digit number whose digit sum is $25$.
6. Write the number "five thousand two hundred and seven" in figures and find its digit sum.
7. Challenge: Try a few rounds of the Kaprekar process from $3215$. Do you reach $6174$?

Activity

Digit-sum check. Pick any five 3-digit numbers from a calendar, a phone book, or your textbook. Find the digit sum of each. Now divide each number by $9$ (using long division) and note the remainder. You should always find that the remainder equals the digit sum modulo $9$ (the remainder when the digit sum is divided by $9$). This is a beautiful, ancient trick.