find the smallest number by which 132 must be multiplied so that the resulting number is a perfect cube
The Correct Answer and Explanation is:
To find the smallest number by which 132 must be multiplied so that the resulting number becomes a perfect cube, we need to analyze the prime factorization of 132.
Step 1: Prime Factorization of 132
132 ÷ 2 = 66
66 ÷ 2 = 33
33 ÷ 3 = 11
11 is a prime number.
So,
132 = 2 × 2 × 3 × 11 = 2² × 3 × 11
Step 2: Check for Cube Requirements
For a number to be a perfect cube, all exponents in its prime factorization must be multiples of 3.
Currently:
- 2 appears with exponent 2
- 3 appears with exponent 1
- 11 appears with exponent 1
To make it a cube:
- We need 1 more 2 (to make 2³)
- We need 2 more 3s (to make 3³)
- We need 2 more 11s (to make 11³)
Step 3: Find the Least Number to Multiply
We need to multiply 132 by the missing factors:
2 × 3² × 11² = 2 × 9 × 121 = 2178
So,
132 × 2178 = a perfect cube
✅ Final Answer: 2178
✍️ Explanation
To transform a number into a perfect cube, all the prime factors in its factorization must have exponents divisible by 3. This is because a perfect cube is a number that can be written in the form x3x^3×3, which means every prime factor of x3x^3×3 is raised to a power that is a multiple of 3.
We begin by factoring 132 into primes:
132 = 2² × 3 × 11.
Now, we examine each exponent:
- The factor 2 appears twice (2²), but we need it to appear three times (2³), so we are missing one 2.
- The factor 3 appears once (3¹), but we need it to appear three times (3³), so we are missing two 3s.
- The factor 11 appears once (11¹), but we need it to appear three times (11³), so we are missing two 11s.
To compensate for the missing primes, we multiply 132 by:
2 × 3 × 3 × 11 × 11 = 2 × 9 × 121 = 2178
When we multiply 132 by 2178, the resulting number has a prime factorization of:
2³ × 3³ × 11³, which is a perfect cube.
Therefore, the smallest number by which 132 must be multiplied to get a perfect cube is 2178.
