We are tasked with determining the value of \( n \) such that the expression \( 2^{3n} - 7n - 1 \) is divisible by a specific number.
To do this, let's first explore the options and substitute various values of \( n \) into the given expression to check divisibility.
We will test each option by substituting the values of \( n \) and calculating \( 2^{3n} - 7n - 1 \).
Step 1: Substitute different values of \( n \)
Start by testing the values \( n = 1, 2, 3, 4, \ldots \), and check when \( 2^{3n} - 7n - 1 \) becomes divisible by 49.
Step 2: Check divisibility
For \( n = 3 \): \[ 2^{3(3)} - 7(3) - 1 = 2^9 - 21 - 1 = 512 - 21 - 1 = 490 \] Clearly, 490 is divisible by 49.
Thus, the correct answer is \( 49 \).
The largest $ n \in \mathbb{N} $ such that $ 3^n $ divides 50! is:
A solid cylinder of mass 2 kg and radius 0.2 m is rotating about its own axis without friction with angular velocity 5 rad/s. A particle of mass 1 kg moving with a velocity of 5 m/s strikes the cylinder and sticks to it as shown in figure.
The angular velocity of the system after the particle sticks to it will be: