Prime with 5 or 6 digits

Number Theory Level 3

How many (positive) prime numbers are there, whose square has 5 or 6 digits?

Details and assumptions

You may use the fact that there are 168 primes less than 1000.

The answer is 143.

9 solutions

Vivek Vicky
Aug 11, 2013

solution: we know that there are 25 prime numbers are between 1-100 and 168 prime numbers less than 1000. 100 x 100=10000(5 digits) 999 x 999=998001(6 digits) 1000 x 1000=1000000(7 digits) so our answer should be same as the number of prime numbers between 100 to 999. hence, 168-25=143. 143 prime numbers will be there less than 1000 whose square has 5 or 6 digits.

Moderator note:

Good clear explanation!

i get it now, . . . the question is wrong that's why i don't get it. . . . the question must be "What prime number whose square is 5 or 6 digits".

Imari Borda - 7 years, 9 months ago

It asks how many not what.

Kushagra Sahni - 5 years, 8 months ago

Your pretty good Vivek I LIKE THE WAY YOU SHOW YOUR solution IS TOO Clear

Red Angel - 7 years, 9 months ago

Alvin Jae Cabato
Aug 13, 2013

Looking at the smallest integer that has a square with 5 digits is 100. The largest integer with a square with 6 digits is 999. Looking at this, we can show that all primes with a square that has 5 or 6 digits is more than 100 and less then 999. In an equation : 100 < x < 999

To make things easier, we can get the number of primes less than 100 and subtract it from the total number of primes less than 1000. This gives us an equation: 168 - 25. The answer to that is 144.

Woops, its supposed to be 143

Alvin Jae Cabato - 7 years, 10 months ago

Joshua Crouch
Aug 12, 2013

997^2=994009 1009^2=1018081

997 is the largest prime number with a square of 5 or 6 digits

101^2=10201 97^2=9409

101 is the smallest prime number with a square of 5 or 6 digits.

101 is the 26th prime number. 997 is the 168th because there are 168 prime numbers below 1000.

168-26+1=143

include<iostream>

include<stdlib.h>

using namespace std; int count=0; void prime(int j) { int flag=0; for(int k=1;k<=j;k++) { if(j%k==0) flag++; }

        if (flag==2)
        {
            if( (j*j)>=10000 && j*j<=999999)
            {
        count++;
        cout<<"Value of j:\t"<<j<<"Square:"<<(j*j)<<"\t count:\t"<<count<<endl;
    }
    }
    }

int main() { int i; for(int j=0;j<=10000;j++) prime(j); cout<<"Value of count\t"<<count<<endl;

// cout<<x<<endl; system("pause"); return 0; }

Vishal Grade - 7 years, 10 months ago

Mayank Dwivedi
Aug 11, 2013

PRIME NUMBERS HAVING SQUARES 5 OR 6 DIGIT ARE STARTING FROM 100. SO,FROM HINTS U CAN FIND THAT ONLY 25 PRIME NUMBERS ARE LESS THAN 100 NOT HAVE SQUARE 5 or 6 DIGITS

Lu Zuokun
Aug 16, 2013

Since it wants prime numbers with sq 5 digits or 6 digits (10000 - 999999), effectively if you sqrt both, you get 100 - 999. Given there are 168 primes under 1000, find the number of primes under 100 (25), then use 168 - 25. This gives us the answer 143.

Aniebiet Akpan
Aug 15, 2013

its very easy, 1-100 have prime squares of at most for digits, 5 and 6 digits are between 101 and 999,. so we get the number of prime squares from 1-100 and subtract from the total number of prime squares, therefore, prime squares of 5-6 digits= total prime squares from 1-1000 minus number of prime squares from 1-100= 168-25=143. simple!

Kevin Kuang
Aug 14, 2013

Given the fact that 168 primes are less than 1000, we know that a number squared, will have the (amount of digits that it has)*2...Therefore the prime must have 3 digits, to determine this number we do 168-25=143, 168 is from total 1-digit, 2-digit and 3-digit primes while 25 is from the number of 1 and 2 digit primes. So our answer here to his question is143

Jose Bohorques
Aug 14, 2013

We know there are 168 primes in the first 1000 numbers

We also know that the square of any number above 99 and below 1000 will have either 5 or 6 digits

So at this point we only have to count the total of prime numbers below 99 which is 25

Therefore:

168 total prime numbers in the first 1000 numbers - 25 prime numbers whose squares have less than 5 digits = 143 prime numbers whose digits are 5 or 6

Devendra Mittal
Aug 14, 2013

there are 25 primes below 100. square(100) = 10000=1st no of 5 digits && square(1000) = 1000000=1st no of 7 digits.1000 is not a prime number so no of prime whose square has 5 digit or 6 digit = no of prime between 100 & 1000. so ans = 168 - 25 = 143