A perfect square is an integer that can be expressed as the product of two equal integers. For example, is a perfect square because it is equal to . If is an integer, then is a perfect square. Because of this definition, perfect squares are always non-negative.
Similarly, a perfect cube is an integer that can be expressed as the product of three equal integers. For example, is a perfect cube because it is equal to Determining if a number is a perfect square, cube, or higher power can be determined from the prime factorization of the number.
Make a list of perfect squares from the smallest.
Thus, the answer is
Find the differences between two adjacent perfect squares, 9 of them from the smallest. For example, start from
Thus, the answers are
Which of the following is not a perfect square?
Since and none of and is the answer. Now, which is not a perfect square but a perfect cube. So, the answer is
In the following equation, and are all distinct positive integers (This is part of the Pythagorean Theorem):
What is the smallest possible value of
Then the answer is
What is the positive number in the following equation:
Then the answer is
What are the perfect squares between and
Then the answers are and
Some properties regarding perfect squares are as follows (their proofs are omitted here):
- Perfect squares cannot have a units digit of 2, 3, or 7. (You can check them out for yourself.)
- The square of an even number is even and the square of an odd number is odd.
- All odd squares are of the form , hence all odd numbers of the form , where is a positive integer, are not perfect squares. For instance, 361 can be written as and we know However, 843 is not a perfect square since and it can be expressed as
- All even numbers of the form , where is a positive integer, are not perfect squares
- All even squares are divisible by 4. (You can take any even square and check this.)
- The difference of 2 odd squares is a multiple of 8. For example, which is
- The sum of the first odd numbers is in fact For example, Here, there are 6 odd numbers, so we can find the sum as just Similarly, as here.
- The sum of the first perfect squares is given by
- If divides then divides as well (Euclid's theorem). From this, we can say that a number is a perfect square if its prime factorization contains all primes raised to some even power.
- Given two positive integers and if is the square of an integer then divides
Ending digits for squared numbers (we consider decimal system):
- If a number has units digit 1 or 9, its square will have units digit 1.
- If a number has units digit 2 or 8, its square will have units digit 4.
- If a number has units digit 3 or 7, its square will have units digit 9.
- If a number has units digit 4 or 6, its square will have units digit 6.
- If a number has units digit 5, its square will have units digit 5.
- If a number has units digit 0, its square will have units digit 0.
The proof for this is left for the reader.
When you cube something, you multiply it by itself three times. For example, . Cubing a positive number will result in a positive number while cubing a negative number will result in a negative number.
Make a list of 10 perfect non-negative cubes starting from the smallest.
Thus the answer is
Some simple properties of perfect cubes are given below. The proofs for them are omitted here.
- Every perfect cube has digital root 1, 8, or 9. By "digital root" we mean the sum of digits that is done until we get a single digit. For instance, the digital root of 1234 can be obtained as follows: We have Since we got a 2-digit number, we add the digits again to get which is the digital root of 1234. The number 54 has digital root Note that if a number has digital root 1, 8, or 9, it does not necessarily mean that it is a perfect cube (as is the case with 54, which has digital root 9 but is not a perfect cube). This can be proven with modular arithmetic.
- Perfect cubes can have any number from 0 to 9 as their units digit.
- The sum of the first perfect cubes is This is equivalent to the square of the sum of the first natural numbers.
- Every positive rational number can be expressed as the sum of three cubes of rational numbers.
- It is possible to express any perfect cube as the sum of four odd numbers. For example,
Units digits of perfect cubes:
- If a number ends in 0, its cube ends in 0.
- If a number ends in 2, its cube ends in 8.
- If a number ends in 3, its cube ends in 7.
- If a number ends in 4, its cube ends in 4.
- If a number ends in 5, its cube ends in 5.
- If a number ends in 6, its cube ends in 6.
- If a number ends in 7, its cube ends in 3.
- If a number ends in 8, its cube ends in 2.
- If a number ends in 9, its cube ends in 9.
The proof for this is left for the reader.
Only 6 of the following 7 numbers are perfect cubes. Which one is not?
By applying the property, the digital root of 40323 is Thus, 40323 is not a perfect cube.
A perfect power is the more general form of squares and cubes. Specifically, it is any number that can be written as the product of some non-negative integer multiplied by itself at least twice. In other words, it is of the form for some integers and
The set of perfect powers is the union of the sets of perfect squares, perfect cubes, perfect fourth powers, and so on. The perfect powers less than or equal to are
A few simple results are given below. The proofs for these are omitted.
- The power of a number with units digit 5 will have units digit 5 again.
- The power of a number with units digit 1 will have units digit 1 again.
- The power of a number with units digit 0 will have units digit 0.
- The power of a number with units digit 6 will have units digit 6.
- The units digit of a number raised to 5 is the units digit of the original number. In fact, where and are positive numbers (which comes from a famous theorem by Euler).
- 0 to the power is 0, where is not equal to zero. 0 to the zeroth power is undefined.
- 1 to the power is 1.
- The number of digits in 10 to the power is where there will be zeroes.
- Every power of a positive integer is a perfect square.
Which is the greatest of the following?
Thus is the greatest among these numbers. Do note that there are better ways to determine which is the greatest or least given a set of numbers; the above powers can all be easily computed.
Find the number of digits in , given and
Since we are given and , we can easily see that the number of digits of any positive integer between 60 and 70 is in fact 13. Here's how: If is greater than then raised to is greater than raised to where and are real numbers and is a positive integer. Knowing this, we can find the number of digits in
Since both and have an equal number of digits and is less than but is less than , must also have an equal number of digits. Hence has 13 digits.