Answer :
Answer:
5
Step-by-step explanation:
To find the smallest number by which 84375 must be multiplied to make the product a perfect cube, we need to perform prime factorization and then determine the necessary adjustments to each prime factor so that the overall product is a perfect cube.
1. **Prime Factorization of 84375:**
- Divide by the smallest prime numbers repeatedly until the quotient is 1.
Let's start with \( 84375 \):
\( 84375 \div 3 = 28125 \) (since \( 8+4+3+7+5 = 27 \), divisible by 3)
\( 28125 \div 3 = 9375 \)
\( 9375 \div 3 = 3125 \)
\( 3125 \div 5 = 625 \)
\( 625 \div 5 = 125 \)
\( 125 \div 5 = 25 \)
\( 25 \div 5 = 5 \)
\( 5 \div 5 = 1 \)
So, the prime factorization is:
\[
84375 = 3^3 \times 5^5
\]
2. **Adjusting the prime factors to form a perfect cube:**
- A perfect cube must have each prime factor raised to a power that is a multiple of 3.
- For \( 3^3 \), it's already a cube.
- For \( 5^5 \), to make it a cube, we need to raise it to the nearest multiple of 3. The next multiple of 3 greater than 5 is 6. Thus, we need an additional \( 5^1 \).
3. **Determine the smallest multiplier:**
- The smallest number by which 84375 must be multiplied to make it a perfect cube is \( 5^1 = 5 \).
4. **Verification:**
- The product is \( 84375 \times 5 = 421875 \).
- Prime factorization of 421875:
\[
421875 = 3^3 \times 5^6
\]
- Both prime factors \( 3^3 \) and \( 5^6 \) are perfect cubes, so 421875 is a perfect cube.
Therefore, the smallest number by which 84375 must be multiplied to make it a perfect cube is \( \boxed{5} \).
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