Three consecutive natural numbers

HOME

Math Central

Quandaries & Queries

Question from abby, a student:

I need help with a problem and it is:

1.what is the smallest four-digit number that is the sum of the three consecutive natural numbers? what are the three consecutive natural numbers that give this sum?

and the second one is:
2.what is the largest three digit number whose square root is a prime?

to who ever answers it -
thank you very much!

Abby,

The smallest four digit number is 1000 so the number you are looking for is larger than 999. Think about the middle number of three consecutive natural numbers. The three consecutive numbers are then

middle number - 1
middle number
middle number + 1

If you add these three numbers you get 3 times the middle number. Hence the number you are looking for is divisible by 3. What is the smallest four-digit natural number that is divisible by 3.

For the second problem suppose the number you are looking for is k, then √k is a prime, call it p. Thus

√k = p

and thus

k = p².

At this point do some arithmetic. Square some primes and see how large p has to be until p² is a four digit number.

Penny

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.