



 
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
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 fourdigit 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
and thus
At this point do some arithmetic. Square some primes and see how large p has to be until p^{2} is a four digit number. Penny
 


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