a 6 digit number is a square as well as a cube.if 6 is subtracted from it a prime no. is obtained.findthe no.

Answer 1

To be a square as well as a cube, the 6 digit number would have to be some number x raised to the power of 6. The only numbers when raised to the 6th power that yield a 6 digit number are:

7^6 = 117649

8^6 = 262144

9^6 = 531441

If you subtract 6 from each of these, you get:

9^6 - 6 = 531441 - 6 = 531435, which is divisible by 5, so it is not prime.

8^6 - 6 = 262144 - 6 = 262138, which is divisible by 2, so it is not prime.

By elimination, that only leaves:

7^6 - 6 = 117649 - 6 = 117643, which we hope is prime. I'm sure you could sure the web to verify that 117643 is prime.

So the 6 digit number you are searching for is 117649.

Hope this helps.

Answer 2

The number is 117649.
To find such a number (let we denote it by x), first observe that the number must be odd. (If the number is even, then if we subtract it by 6 will produce an even number that can't be a prime number. The only even prime number is 2). It is clear that x is positive.
Suppose that x = y^3 = z^2. Then y and z must be odd. The possible last digit for x is 1,3,5,7, and 9. Since there are no integer which square has the last digit 3 and 7, then the possible last digit for x reduce to 1,5, and 9. Now suppose that the last digit for x is 1. Then if we subtract x by 6 it will produce a number which last digit is 5. Clearly this number is not a prime number. Then x will last either by 5 or 9.
Now observe that the possible last digit for y is 5 and 9. The possible value for y is 55, 65, 75, 85, 95, 49, 59, 69, 79, 89, and 99. You can calculate the cube of those values and take the square root of it. The only value that give an integer is y = 49. Then x = y^3 = 117649 = 343^2.
Subtract x by 6, we will have the number 117643, which can be easily proved to be a prime number.

Answer 3

Since the number is both a square and a cube, then it must be a sixth power. By taking the sixth root of 100000 and rounding up, we see the smallest integer that produces a six-digit power is 7. In fact, 7, 8, and 9 are the only numbers whose sixth power is a six-digit number. 7^6 = 117649. 117649 - 6 = 117643, which is prime.

Answer 4

AAAAAAAAAAAHHHHHHHHHH!!!!

I just spent the last 30 minutes entering squares into my calculator. I would subtract 6 to see if prime, and then try the cube root. I finally reached 117649, 343 squared, 49 cubed, and prime number 117643 + 6. Please pick me for best answer after all that work!!!

Answer 5

117649