Understanding and Calculating the Smallest Positive Integer ( n ) Such That ( n^3 ) is Divisible by 2020 Without Remainders
Introduction to Prime Factorization
To understand the problem, we first need to break down the number 2020 into its prime factors. The prime factorization of 2020 is as follows:
Prime Factorization of 2020
The prime factors of 2020 are 22 5 101. This means that 2020 can be expressed as:
2020 22 5 101
Given this, we need to find the smallest positive integer ( n ) such that ( n^3 ) is divisible by 2020 without any remainders. This problem requires a deep understanding of divisibility and prime factorization.
Breaking Down the Factors for ( n^3 )
To ensure that ( n^3 ) is divisible by 2020, ( n ) itself must contain at least the prime factors 2, 5, and 101 in sufficient numbers to make ( n^3 ) divisible by 2020.
Let’s consider the prime factorization again:
Required Prime Factors for ( n )
The smallest ( n ) must include:
At least one 2 At least one 5 At least one 101Hence, the smallest ( n ) would be the product of these prime factors,
Thus, the smallest ( n ) is:
2 5 101 1010
And this ( n ) will ensure that ( n^3 ) contains:
23 (since ( 1010^3 ) includes 101sup3; and 1010 contains 22) 53 1013Therefore:
1010sup3; 23 53 1013 2020 510050
Conceptualizing the Smallest Cube Divisible by 2020
Alternatively, we can also deduce the smallest cube divisible by 2020. For this, we will round up the factors in the prime factorization to the nearest multiple of 3 to ensure divisibility by 3. Let's do this step by step:
2020 22 5 101
For ( n^3 ) to be divisible by 2020, the prime factorization of ( n ) must include:
23 (since 22 rounds up to 23) 53 1013Thus, the smallest ( n ) is:
23 53 1013 10203 1030301000
Hence, the smallest ( n ) such that ( n^3 ) is divisible by 2020 is:
1030301000
Conclusion
The smallest positive integer ( n ) such that ( n^3 ) is divisible by 2020 without any remainders is:
1010
This solution is derived from understanding the prime factorization of 2020 and ensuring that the smallest ( n ) contains sufficient prime factors to make ( n^3 ) divisible by 2020. The process involves breaking down the problem using prime factorization, rounding up the factors to the nearest multiple of 3, and ensuring that ( n ) includes all necessary prime factors.