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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
Is 333 a a cube number?
No, 333 is not a cube number. A cube number is an integer that can be expressed as the cube of another integer, such as (1^3 = 1), (2^3 = 8), or (3^3 = 27). The cube root of 333 is approximately 6.9, which is not an integer, indicating that 333 cannot be expressed as (n^3) for any integer (n).
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
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
Is 333 a a cube number?
No, 333 is not a cube number. A cube number is an integer that can be expressed as the cube of another integer, such as (1^3 = 1), (2^3 = 8), or (3^3 = 27). The cube root of 333 is approximately 6.9, which is not an integer, indicating that 333 cannot be expressed as (n^3) for any integer (n).
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.
Is 83 a cube number?
No, 83 is not a cube number. A cube number is an integer that can be expressed as the cube of another integer (n^3). The cube numbers closest to 83 are 64 (4^3) and 125 (5^3), indicating that 83 falls between these two values and is not itself a cube.
Can the cube root of a positive number ever be a negative?
No, the cube root of a positive number cannot be negative. The cube root of a positive number is also positive, as multiplying three positive factors will always yield a positive result. In contrast, the cube root of a negative number is negative, but this does not apply to positive numbers.
