Find the smallest number by which 375 must be divided to obtain a perfect cube, please.
The Correct Answer and Explanation is:
To find the smallest number by which 375 must be divided to obtain a perfect cube, let’s start by factoring 375 into its prime factors.
Step 1: Prime Factorization of 375
375 can be divided by 5, as it ends in 5.375÷5=75375 \div 5 = 75375÷5=75
Next, divide 75 by 5:75÷5=1575 \div 5 = 1575÷5=15
Finally, divide 15 by 5:15÷5=315 \div 5 = 315÷5=3
So, the prime factorization of 375 is:375=53×3375 = 5^3 \times 3375=53×3
Step 2: Identifying What Needs to Be Divided
To make 375 a perfect cube, each prime factor’s exponent must be a multiple of 3. In the prime factorization 53×35^3 \times 353×3, the exponent of 5 is already a multiple of 3 (since 535^353), but the exponent of 3 is 1, which is not a multiple of 3.
Step 3: Making 375 a Perfect Cube
To make 375 a perfect cube, we need to divide by a number that removes the factor of 333 (since it appears to the first power and should be removed for it to be a perfect cube). This means the smallest number we need to divide by is 3.
Step 4: Final Calculation
Now, divide 375 by 3:375÷3=125375 \div 3 = 125375÷3=125
Check if 125 is a perfect cube:125=53125 = 5^3125=53
Since 125=53125 = 5^3125=53, it is indeed a perfect cube.
Conclusion
The smallest number by which 375 must be divided to obtain a perfect cube is 3.
