Finding the Smallest Natural Number
We need to find the smallest natural number, let's call it $N$, such that when $N$ is divided by 20, 42, or 76, the remainder is always 7.
This can be expressed mathematically as:
- $N \equiv 7 \pmod{20}$
- $N \equiv 7 \pmod{42}$
- $N \equiv 7 \pmod{76}$
This implies that $N - 7$ must be exactly divisible by 20, 42, and 76. Therefore, $N - 7$ must be a common multiple of these three numbers.
To find the smallest such natural number $N$, we need $N - 7$ to be the least common multiple (LCM) of 20, 42, and 76.
Calculating the LCM
First, find the prime factorization of each number:
- $20 = 2^2 \times 5$
- $42 = 2 \times 3 \times 7$
- $76 = 2^2 \times 19$
The LCM is found by taking the highest power of each prime factor present in any factorization:
$\text{LCM}(20, 42, 76) = 2^2 \times 3 \times 5 \times 7 \times 19$
Calculating the value:
$\text{LCM}(20, 42, 76) = 4 \times 3 \times 5 \times 7 \times 19 = 12 \times 5 \times 7 \times 19 = 60 \times 7 \times 19 = 420 \times 19 = 7980$
Determining the Number N
Now we know that $N - 7 = \text{LCM}(20, 42, 76)$:
$N - 7 = 7980$
Solving for $N$:
$N = 7980 + 7$
$N = 7987$
Thus, the smallest natural number that leaves a remainder of 7 when divided by 20, 42, or 76 is 7987.
Important Questions from Numerical Computation
$A$ is an ($n \times n$) matrix. Consider the following two statements
Statement 1: Columns of matrix $A$ are linearly independent
Statement 2: Inverse of matrix $A$ exists
Which one of the following statements is TRUE?
- Consider the function $G(x, y, z) = 0$. This function allows us to implicitly define each of three variables as a function of the other two variables. Assume that all partial derivatives of the function $G(x, y, z)$ exist everywhere. Then, the value of $\left(\frac{\partial z}{\partial x} \times \frac{\partial x}{\partial y} \times \frac{\partial y}{\partial z}\right)$ is _________ (in integer)
- Levenshtein distance is used to measure the minimum edit distance between two strings by counting the minimum number of editing operations (such as, insertions, deletions, substitutions) required to transform one string to another. Consider an alternative version of the Levenshtein distance in which each insertion and each deletion has a cost of 1 and substitutions are not allowed.
Based on this alternative version, the Levenshtein distance between the strings, word and work is ______ (Answer in integer). - In a given collection of documents, let $N$ be the total number of documents and let $d$ be the number of documents in which the term $t$ occurs.
Which ONE of the following fractions is used to define the inverse document frequency (idf) of the term $t$? - The total number of unique character bigrams in the string thoughtsoftheghosts (which contains 19 characters) is ______.
