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
- An organization allows its employees to work independently on consultancy projects but charges an overhead on the consulting fee. The overhead is 20% of the consulting fee, if the fee is up to . 5,00,000. For higher fees, the overhead is . 1,00,000 plus 10% of the amount by which the fee exceeds . 5,00,000. The government charges a Goods and Services Tax of 18% on the total amount (the consulting fee plus the overhead). An employee of the organization charges this entire amount, i.e., the consulting fee, overhead, and tax, to the client. If the client cannot pay more than . 10,00,000, what is the maximum consulting fee that the employee can charge?
Three frictionless pulleys with rope attachment are in a static equilibrium as shown in the figure. The mass $m_1$ and $m_2$, in kg, respectively are
If the in-situ density of coal is 1320 kg/m$^3$ and the density of blasted coal is 952 kg/m$^3$, the swell factor is _____________ (rounded off to 3 decimal places)
A five-member truss system is shown in the figure. The maximum vertical force P in kN that can be applied so that loads on the member CD and BC do NOT exceed 50 kN and 30 kN, respectively is _____________(rounded off to 2 decimal places)
The Fourier transform and its inverse transform are respectively defined as $\tilde{f}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(x)e^{i\omega x}dx$ and $f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \tilde{f}(\omega)e^{-i\omega x}d\omega$. Consider two functions $f$ and $g$. Another function $f * g$ is defined as
$(f * g)(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} f(y)g(x - y)dy$
Which of the following relation is/are true?
Note: Tilde ($\sim$) denotes the Fourier transform.
