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To find the least positive integer ( k ) for which ( 15k ) is a cube, we start with the prime factorization of 15, which is ( 3^1 \times 5^1 ). For ( 15k ) to be a perfect cube, the exponents in its prime factorization must be multiples of 3. Thus, we need to make the exponents of both 3 and 5 in ( 15k ) equal to 3. Therefore, ( k ) must contribute ( 3^2 ) (to make the exponent of 3 equal to 3) and ( 5^2 ) (to make the exponent of 5 equal to 3). Thus, ( k = 3^2 \times 5^2 = 9 \times 25 = 225 ). Therefore, the least positive integer ( k ) is ( 225 ).
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What is the least positive integer x for which 12x is the cube of an integer?
2
What is the least positive integer n such that the product 800n is a perfect cube?
10
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
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.
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.
What is the least positive integer x for which 12x is the cube of an integer?
2
What is the least positive integer x for which 594x is the cube of an integer?
x = 484
What is the least positive integer n such that the product 800n is a perfect cube?
10
What is the least positive integer a for which 25a is the cube integer?
The only solution is that a = 5 then 25a = 25 x 5 = 52 x 5 = 53.
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
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.
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 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.
Explain with workings What is the smallest positive integer q such that 25q is a perfect cube?
5
Is 30 a perfect cube?
No, 30 is not a perfect cube. A perfect cube is a number that can be expressed as the cube of an integer, such as (1^3 = 1), (2^3 = 8), or (3^3 = 27). The cube root of 30 is approximately 3.11, which is not an integer, confirming that 30 cannot be written as the cube of any whole number.
Is 5 a cube number?
No, 5 is not a cube number. A cube number is the result of an integer multiplied by itself twice (n × n × n). The cube numbers closest to 5 are 1 (1³) and 8 (2³).
