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Does a by b plus b by c plus c by a equals 3 imply that abc is the cube of an integer when a b c are integers?
No. Here is a proof by counterexample that it does not.
Given ab + bc + ca = 3:
Assume toward a contradiction that abc is a cube. Then a = b = c.
Without loss of generality, let a = 2, b = 2, and c = 2.
Then ab = 4, bc = 4, and ca = 4.
ab + bc + ca = 4 + 4 + 4 = 12.
Therefore, 12 = 3, which is false, and so the original statement is false.
What else can I help you with?
How does finding cube root differ from finding a square root of a positive integer?
Finding the square root of a positive integer involves identifying a number that, when multiplied by itself, equals the original integer, resulting in one non-negative solution. In contrast, finding the cube root of a positive integer determines a number that, when multiplied by itself twice (i.e., raised to the power of three), equals the original integer, yielding one real solution. The key difference lies in the operations involved: square roots deal with pairs of factors, while cube roots involve triplets. Additionally, cube roots can yield real solutions for negative integers, unlike square roots.
Is 121 a cube number?
No, 121 is not a cube number. A cube number is the result of an integer multiplied by itself three times (n × n × n). The cube of integers around the square root of 121 (10) are 8 (2^3) and 27 (3^3), neither of which equals 121. Thus, 121 is classified as a square number (11 × 11), not a cube number.
Is 11025 a perfect cube?
No, 11025 is not a perfect cube. A perfect cube is an integer that can be expressed as the cube of another integer. The cube root of 11025 is approximately 22.2, which is not an integer, indicating that 11025 cannot be written as ( n^3 ) for any integer ( n ).
The square root of a perfect square and the cube root of a perfect cube is?
an integer
Why is 343 a perfect cube?
343 is a perfect cube because it can be expressed as (7^3) (7 multiplied by itself three times: (7 \times 7 \times 7 = 343)). In mathematics, a perfect cube is a number that can be formed by raising an integer to the third power. Since 7 is an integer and its cube equals 343, this confirms that 343 is indeed a perfect cube.
How does finding cube root differ from finding a square root of a positive integer?
Finding the square root of a positive integer involves identifying a number that, when multiplied by itself, equals the original integer, resulting in one non-negative solution. In contrast, finding the cube root of a positive integer determines a number that, when multiplied by itself twice (i.e., raised to the power of three), equals the original integer, yielding one real solution. The key difference lies in the operations involved: square roots deal with pairs of factors, while cube roots involve triplets. Additionally, cube roots can yield real solutions for negative integers, unlike square roots.
Is 11025 a perfect cube?
No, 11025 is not a perfect cube. A perfect cube is an integer that can be expressed as the cube of another integer. The cube root of 11025 is approximately 22.2, which is not an integer, indicating that 11025 cannot be written as ( n^3 ) for any integer ( n ).
The square root of a perfect square and the cube root of a perfect cube is?
an integer
What is the smallest integer where the integer before it is a perfect square and the integer after it is a perfect cube?
The integer is 26
Is 77 a cube number?
If by cube you mean perfect cube (a cube of an integer), then no, and the nearest perfect cube is 81.
How is 784 a perfect cube?
The number 784 is not a perfect cube. A perfect cube is defined as a number that can be expressed as the cube of an integer (i.e., (n^3) for some integer (n)). The cube root of 784 is approximately 9.24, which is not an integer, indicating that 784 cannot be represented as a whole number cubed.
What is the least positive integer x for which 12x is the cube of an integer?
2
Why are perfect square and cube perfect?
Because they are square/cube of an integer.
What is the difference between a cube and a perfect cube?
A cube is any number multiplied by itself three times, eg 2 cubed = 2³ = 2×2×2 = 8; 1.5³ = 1.5×1.5×1.5 = 3.375 A perfect cube is an integer (whole number) that is the cube of an integer, eg 8 is a perfect cube as it is 2 cubed, but 9 is not a perfect cube as 9 = 2.08008382...³
What is the smallest positive integer e such that the cube root of 600e is an integer?
45
What is the least positive integer x for which 594x is the cube of an integer?
x = 484
What is the smallest possible value of k if k is a perfect cube?
The smallest possible value of ( k ) if ( k ) is a perfect cube is ( 1 ), since ( 1^3 = 1 ). Perfect cubes are formed by multiplying integers by themselves three times, and the smallest integer, which is ( 1 ), yields the smallest perfect cube. Therefore, ( k = 1 ) is the answer.
