How many positive integers less than or equal to 4000 can be
written down without using the digits 7, 8 or 9?
Consider all two digit numbers (10, 11, . . . ,99). In writing down
all these numbers, which digits occur least often, and which occur
most often ? What about three digit numbers, four digit numbers and
How many six digit numbers are there which DO NOT contain a 5?
Published September 2000,February 2011.
Every credit card is uniquely identified by a 16-digit number.
The first 15 digits are assigned by the bank issuing the card. The
last digit (called the check digit) is determined by a formula
which enables a computer to check that the number has been
correctly entered when the card is used in a transaction.
All the major credit card companies use a system called Codabar
to assign the check digit. This is how it works.
1. Suppose that the first 15 digits are
$3141 5926 5358 979$
2. Add the digits in the odd positions and double the total.
3. Add the digits in the even positions.
4. Count the number of digits in the odd positions that are
greater than $4$. The numbers in this example are $5$, $5$, $5$,
$9$ and $9$, that is altogether $5$ numbers.
5. Now add together the three numbers obtained in this way to
6. The check digit is the number needed to bring this total up
to the next multiple of $10$. What should this be in our example?
Yes, the required check digit is $6$.
The Codabar system is one of the most efficient error detection
methods. It picks up 100% of all single digit errors, and most
other common errors such as switching two adjacent digits.
Have a look at some credit cards for yourself and you will find
that the last digit is given by the formula described here. Test it
to see if it picks up errors likely to be made in transcribing the