Smallest Multiplier Divisor For Perfect Cubes And True False Statements

In this article, we will explore two interesting mathematical problems related to perfect cubes. First, we'll determine the smallest number by which 2560 must be multiplied to obtain a perfect cube. Second, we'll find the smallest number by which 8788 must be divided to achieve a perfect cube. Additionally, we will evaluate the truthfulness of a given statement about the divisibility of 650. Let's delve into the world of numbers and perfect cubes!

Finding the Smallest Multiplier for a Perfect Cube

To find the smallest number by which 2560 must be multiplied to make the product a perfect cube, we'll embark on a journey of prime factorization. Prime factorization is the bedrock of this process, allowing us to dissect the number into its fundamental building blocks. Let's break down 2560 into its prime factors:

2560 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 5 = 2^9 × 5^1

Now, let's understand what constitutes a perfect cube. A perfect cube is a number that can be expressed as the product of an integer with itself three times. In terms of prime factors, this means each prime factor must appear a multiple of three times (i.e., its exponent must be divisible by 3). Looking at the prime factorization of 2560, we notice that the exponent of 2 (which is 9) is divisible by 3, but the exponent of 5 (which is 1) is not. To make 2560 a perfect cube, we need to make the exponent of 5 a multiple of 3.

To achieve this, we must multiply 2560 by a number that introduces two more factors of 5. This will raise the exponent of 5 to 3, which is a multiple of 3. Therefore, the smallest number by which 2560 must be multiplied is 5 × 5 = 25.

When we multiply 2560 by 25, we get:

2560 × 25 = 2^9 × 5^1 × 5^2 = 2^9 × 5^3 = (2³⁾3 × 5^3 = (2^3 × 5)^3 = (8 × 5)^3 = 40^3 = 64000

As you can see, 64000 is a perfect cube (40 × 40 × 40), confirming that 25 is indeed the smallest multiplier.

In conclusion, the smallest number by which 2560 must be multiplied to obtain a perfect cube is 25. This process of prime factorization and analyzing exponents is crucial in solving problems related to perfect squares, perfect cubes, and other similar concepts.

Finding the Smallest Divisor for a Perfect Cube

Now, let's tackle the second part of our exploration: finding the smallest number by which 8788 must be divided to yield a perfect cube. This problem, similar to the previous one, hinges on the concept of prime factorization and understanding the structure of perfect cubes.

We begin by decomposing 8788 into its prime factors:

8788 = 2 × 2 × 13 × 13 × 13 = 2^2 × 13^3

Recall that a perfect cube requires each of its prime factors to have an exponent that is a multiple of 3. Examining the prime factorization of 8788, we observe that the exponent of 13 is already 3, which is a multiple of 3. However, the exponent of 2 is 2, which is not a multiple of 3. To transform the result of the division into a perfect cube, we need to eliminate the extra factors of 2 that prevent the exponent from being a multiple of 3.

In this case, we have 2^2, meaning we have two factors of 2. To make the exponent of 2 a multiple of 3 (specifically, 0), we need to divide by 2^2, which is 4. Dividing 8788 by 4 will effectively remove these extra factors of 2.

Let's perform the division:

8788 ÷ 4 = (2^2 × 13^3) ÷ 2^2 = 13^3 = 2197

We find that 2197 is indeed a perfect cube, as 2197 = 13 × 13 × 13 = 13^3. This confirms that 4 is the smallest number by which 8788 must be divided to obtain a perfect cube.

Therefore, the answer to our second problem is 4. This method of identifying prime factors and adjusting their exponents through division (or multiplication, as we saw earlier) is a powerful technique in number theory.

True or False Statement Evaluation

Finally, let's address the true/false statement: "650 is not a..." (The statement is incomplete in the original prompt, so I will address it with a few possibilities). This part of our exploration delves into the realm of divisibility and number properties.

To determine the truthfulness of the statement, we need to understand what property the statement refers to. Since the statement is incomplete, let's analyze a few possibilities:

Possibility 1: 650 is not a perfect square.

To check if 650 is a perfect square, we need to see if its square root is an integer. The square root of 650 is approximately 25.495, which is not an integer. Alternatively, we can perform prime factorization:

650 = 2 × 5 × 5 × 13 = 2 × 5^2 × 13

For a number to be a perfect square, all the exponents in its prime factorization must be even. In this case, the exponents of 2 and 13 are 1, which is odd. Therefore, 650 is not a perfect square. The statement "650 is not a perfect square" is True.

Possibility 2: 650 is not a perfect cube.

Similar to the perfect square check, we can find the cube root of 650, which is approximately 8.66. This is not an integer, suggesting 650 is not a perfect cube. Looking at the prime factorization (650 = 2 × 5^2 × 13), we see that none of the exponents are multiples of 3. Thus, 650 is not a perfect cube. The statement "650 is not a perfect cube" is True.

Possibility 3: 650 is not divisible by 3.

A quick way to check divisibility by 3 is to sum the digits of the number. If the sum is divisible by 3, the number is also divisible by 3. For 650, the sum of the digits is 6 + 5 + 0 = 11, which is not divisible by 3. Therefore, 650 is not divisible by 3. The statement "650 is not divisible by 3" is True.

Possibility 4: 650 is not a prime number.

A prime number has exactly two distinct positive divisors: 1 and itself. 650 has several divisors (1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, and 650), so it is not a prime number. The statement "650 is not a prime number" is True.

Without the complete statement, we can conclude that for several common properties, the statement "650 is not a..." is true. This demonstrates the importance of understanding divisibility rules and number classifications.

In this article, we successfully determined the smallest multipliers and divisors required to transform numbers into perfect cubes. We leveraged the power of prime factorization to dissect numbers and analyze their exponents. Furthermore, we evaluated the truthfulness of a statement about 650, exploring divisibility and number properties. These exercises highlight the beauty and practicality of number theory in problem-solving and mathematical reasoning. Mastering these concepts builds a strong foundation for tackling more advanced mathematical challenges.