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Chapter 8: Problem 14

Find the number of positive integers not exceeding 1000 that are either the square or the cube of an integer.

Short Answer

Expert verified
There are 38 positive integers not exceeding 1000 that are either squares or cubes of an integer.

Step by step solution

01

Identify squares and cubes

Identify the positive integers not exceeding 1000 that are squares or cubes of integers. Let us start by finding the squares.
02

Calculate perfect squares

Find the maximum integer whose square is less than or equal to 1000, which is \(\floor{\sqrt{1000}}\). That gives 31, since \(\floor{\sqrt{1000}}\) = 31. The perfect squares are 1, 4, 9, 16, ..., 961.
03

Calculate perfect cubes

Find the maximum integer whose cube is less than or equal to 1000, which is \(\floor{\sqrt[3]{1000}}\). That gives 10, since \(\floor{\sqrt[3]{1000}}\) = 10. The perfect cubes are 1, 8, 27, ..., 1000.
04

Calculate sixth powers

Identify numbers that are both squares and cubes (i.e., sixth powers). The integers whose sixth power is less than or equal to 1000 are given by \( \floor{\sqrt[6]{1000}} \). That gives 3, since \( \floor{\sqrt[6]{1000}} \) = 3. The sixth powers are 1, 64, 729.
05

Count the numbers

There are 31 perfect squares and 10 perfect cubes. Subtract the sixth powers counted twice. There are 3 sixth powers. The total count is \( 31 + 10 - 3 = 38 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perfect Squares
When we talk about perfect squares, we're looking at numbers that can be the result of squaring a whole number. For example, starting with 1 and going through 31, the numbers that fit this are 1, 4, 9, 16, and so on, all the way to 961.

A perfect square is formed by multiplying an integer by itself. Mathematically, if x is an integer, then x虏 is known as a perfect square. For instance:
  • 1虏 = 1
  • 2虏 = 4
  • 3虏 = 9
  • And so forth...
By calculating the number of perfect squares up to 1000, we can find that the highest number we reach is 31虏, which is 961. Therefore, there are 31 perfect squares under 1000.
Perfect Cubes
A perfect cube is formed by taking an integer and raising it to the power of 3. This means multiplying the number by itself three times. For example:
  • 1鲁 = 1
  • 2鲁 = 8
  • 3鲁 = 27
  • ... up to 10鲁 = 1000
A perfect cube right below 1000 is 10鲁. Therefore, the perfect cubes under 1000 are 1, 8, 27, and go up to 1000.

Understanding perfect cubes is pretty straightforward: just keep in mind that it's about multiplying the same number three times. This means there are 10 perfect cubes up to 1000.
Sixth Powers
Numbers that are both perfect squares and perfect cubes are called sixth powers. This is because raising a number to the power of 6 means squaring it and then cubing it (or vice versa).

For example, 1, 2, and 3 raised to the sixth power are:
  • 1鈦� = 1
  • 2鈦� = 64
  • 3鈦� = 729
These numbers are both perfect squares and perfect cubes simultaneously.

The sixth powers less than 1000 are 1, 64, and 729. There are three sixth powers in the specified range.

Understanding these concepts helps simplify the original problem, ensuring that sixth powers are not counted multiple times when adding perfect squares and cubes.

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Most popular questions from this chapter

Write out the explicit formula given by the principle of inclusion-exclusion for the number of elements in the union of five sets.

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