[Math Lair] Niven Numbers

Math Lair Home > Topics > Niven Numbers

A Niven number is a number that is divisible by the sum of its digits. For example, 108 is a Niven number because 1 + 0 + 8 = 9 and 108 is divisible by 9 (many, but not all, multiples of 9 are Niven numbers). On the other hand, 121 is not a Niven number because 1 + 2 + 1 is 4 and 121 is not divisible by 4. These are also called Harshad numbers.

An interesting exercise for students is to determine how many Niven numbers there are in a given range (say, between 100 and 200). There are many shortcuts that can be used to determine the answer without having to perform the division for every number. For example, divisibility tests can come in handy here. Being aware of the properties of even and odd numbers can also save time. Because an odd number cannot be the product of an even number and another number, there's no need to attempt to divide 121 by 4, or 123 by 6, etc. to try to determine if it's a Niven number of not; because odd numbers cannot have even factors, 121 cannot be a Niven number.

An interesting diversion is finding consecutive numbers that are Niven numbers. Ignoring the numbers 1 through 9, the first sequence of three consecutive numbers that are all Niven numbers are 110, 111, and 112. The first such sequence of four numbers is 510 through 513, and the first such sequence of five numbers is 131,052 through 131,056. C. Cooper and R. E. Kennedy have shown that there can be at most 20 consecutive Niven numbers. There are an infinite number of such sequences, but the numbers in the smallest such sequence have 44,363,342,786 digits!

Niven numbers can be generalized to different bases as well. For example, 300 in base 10 is a Niven number, because 300 is divisible by 7. In base 7, 30010 = 6067, and 6067 is also a Niven number, because 30010 is divisible by 6 + 0 + 6 = 12. Cooper and Kennedy's result above can be generalized in that there can be at most 2n consecutive Niven numbers in base n. The largest number that is a Niven number in every base is 6.