How many positive integers <= 1000000 are squares or cubes?

Hi,

I have been set some weird questions.

1. How many positive integers <= 1000000 are squares or cubes?

...

2. How many positive integers <=1000 include among their (base 10) digits a "5" or a "6"

Take the square root of 1000000 to give the number of integers whose square is <= 1000000.
Similarly for the cubed integers.

Take the square root of 1000000 to give the number of integers whose square is <= 1000000.
Similarly for the cubed integers.

3. How many solutions in positive integers are there to the equation
x + y + z = 2004?

How would the square root iof 1000000 tell you the no. of integers that square <= 1000000? Is there like a proof?

How would the square root iof 1000000 tell you the no. of integers that square <= 1000000? Is there like a proof?

Think about it - if you sqrt (or whatever-root) a number, all numbers less than the root will square (or whatever) to be less than the original number. Try it with 100: it's obvious that there are 10 integers whose squares are <= 100. 10, and any integer less than 10.

Ben

But couldn't there be a cube which is also a square? (eg. 1000000=1000^2=100^3). Surely you'd be counting these cases twice.

5,6,15,16,25,26....95,96 (8x2x8)

Giving 200+160+128=488 positive integers with a 5 or 6 in their digits.

Yes, but the question said, " How many positive integers <= 1000000 are squares or cubes?"
Perhaps it's a matter of interpretation. I took that to mean the number that were squares and the number that were cubes.
Hmm, anyone else got a comment?

how many integers under 1x10^6 are squares and cubes?

why is it not just 8x2
5,6,15,16,25,26,35,36,45,46,75,76,85,86,95,96

10 integers under(and including 10^6 itself) are squares and cubes, ie 6th powers.
If you mean squares or cubes, then its
sqrt(10^6) + cuberoot(10^6) - 6th root(10^6)

Yeah, I just got around to working that out
Here's my version.

N = 1,000,000
n1 = &#8730;N = 1000 => there are 1000 numbers, the integers for 1 to 1000 inclusive, that have squares less than, or equal to, 1,000,000.
n2 = ³&#8730;N = 100 => there are 100 numbers, the integers for 1 to 100 inclusive, that have cubes less than, or equal to, 1,000,000.

let n,m be integers s.t. n² <= 1,000,000 and m³ <= 1,000,000.
n and m are constrained by having n=k³ and m=k²
=> k^6 = 1,000,000
=> k<=10
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In other words, all the integers form 1 to 10 inclusive can be taken to the sixth power and be less than or equal to 10^6.
And each one of these numbers, to the sixth power, can have a square root and a cube rooot.
So there are 10 duplicates in our list of squares and cubes.Total number of integers <= 1,000,000 which are squares or cubes is,

n = n1 + n2 - 10
n = 1000 + 100 - 10
n = 1090
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