Understanding the Problem:
We need to find the smallest number by which 375 must be multiplied to obtain a perfect cube.

Method:
To solve the problem, we use the prime factorization method. We will find the prime factors of 375, and then we will group them in triplets. Each triplet will be multiplied to obtain a perfect cube. Finally, we will take the product of all these triplets to get the smallest number by which 375 must be multiplied to obtain a perfect cube.

Prime Factorization of 375:
To find the prime factors of 375, we will use the division method.

375 ÷ 5 = 75
75 ÷ 5 = 15
15 ÷ 3 = 5

So, the prime factors of 375 are 3, 5, and 5.

Grouping the Prime Factors:
Now, we will group the prime factors of 375 in triplets.

3 × 5 × 5 = 75 (a perfect cube)
So, one triplet is 3 × 5 × 5.

Calculating the Smallest Number:
Finally, we will take the product of all these triplets.

3 × 5 × 5 = 75 (one triplet)
3 × 5 × 5 = 75 (second triplet)
3 × 5 × 5 = 75 (third triplet)

Product of all the triplets = 75 × 75 × 75 = 421875

Therefore, the smallest number by which 375 must be multiplied to obtain a perfect cube is 421875.

Conclusion:
We used the prime factorization method to find the smallest number by which 375 must be multiplied to obtain a perfect cube. We found that the prime factors of 375 are 3, 5, and 5. We grouped these prime factors in triplets and calculated the product of all these triplets to get the smallest number, which is 421875.