Tom now had access to the ammunition [ammo]. He now had to load a correct amount of ammo to shoot the combatant. The gun calculated the amount of ammo to be loaded and displayed the following to Tom –

“There are 27 bullets that can be fired in each roundand 64 rounds are available, which means $27 \times 64 = 2^{x} \times 3^{y}$ bullets can be shot”

The gun can be activated digitally by entering another passcode, which is the numerical value of (x-y).

What is the passcode that Tom has to enter? ____ [Enter a numerical value]

_

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Solution

We know that $27 = 3^{3}$
Also $64 = 8^{2} = {(2^{3})}^{2} = 2^{(3 \times 2)} = 2^{6}$
Now, $27 \times 64 = 3^{3} \times 2^{6}$

Comparing it with $2^{x} \times 3^{y}$ , we get $x = 6$ and $y = 3$

x-y = 6-3 =3

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Tom now had access to the ammunition [ammo]. He now had to load a correct amount of ammo to shoot the combatant. The gun calculated the amount of ammo to be loaded and displayed the following to Tom –

“There are 27 bullets that can be fired in each roundand 64 rounds are available, which means $27 \times 64 = 2^{x} \times 3^{y}$ bullets can be shot”

The gun can be activated digitally by entering another passcode, which is the numerical value of (x-y).

What is the passcode that Tom has to enter? ______ [Enter a numerical value]

___

Having entered the passcode correctly, Tom got his mission. He needs to fight an enemy combatant and get information about the next clue. Tom found the room where the combatant was, but to enter the room, he needed another passcode.The passcode is the answer to the following display:

“The lock thinks that 0.36 = 3 $\times$ $10^{a}$ + 6 $\times$ $10^{b}$ and the passcode is the value (a $\times$ b)”

What is the passcode that Tom has to enter? ______ [Enter a numerical value]

___

Once Tom answered in and drove off to reach his destination – a building calledTheArchitecture.This building closely resembles a Rubik’s Cube, in the sense that the “rooms” of The Architecture are like the “cubes” of the Rubik’s Cube, as shown:

This building has 3 floors and each floor has 3 $\times$ 3 [read 3 by 3 – which means there are 3 rows and 3 columns] rooms which are all identical in size and shape.Tom stopped the car in front of the building.To enter this architecture, he should enter a passcode which is equal to the total number of rooms [cubes] in the building.

What is the passcode that Tom has to enter? ______ [Enter a numerical value]

___

With sufficient ammo loaded, Tom managed to overpower his combatant and got the next clue and exited The Architecture.

Following the clue, Tom reached a particular structure called Exponere Quattuor. He had to go inside this structure for which he had to get past the entry.A commando was guarding the entry. Tom got a message from his spies that the only way to get past the commando is to give him the code which was – the minimum value of x (integer) for which $3^{2} < (- 3)^{x}$ .

The code Tom gave was ____ [Enter a numerical value].

___

Once Tom answered the question to unlock the car, he hopped in and drove off to reach his destination – a building calledTheArchitecture.This building closelyresembles a Rubik’s Cube, in the sense that the “rooms” of The Architectureare like the “cubes” of the Rubik’s Cube, as shown:

This building has 3 floors and each floor has 3 $\times$ 3 [read 3 by 3 – which means there are 3 rows and 3 columns] rooms which are all identical in size and shape.Tom stopped the car in front of the building.To enter this architecture, he should enterapasscodewhich is equal to the total number of rooms [cubes] in the building.

What is the passcode that Tom has to enter? ______ [Enter a numerical value]

___

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We know that 27=33 Also 64=82=(23)2=2(3×2)=26 Now, 27×64=33×26 Comparing it with 2x×3y, we get x=6 and y=3 x-y = 6-3 =3

We know that $27 = 3^{3}$
Also $64 = 8^{2} = {(2^{3})}^{2} = 2^{(3 \times 2)} = 2^{6}$
Now, $27 \times 64 = 3^{3} \times 2^{6}$

Comparing it with $2^{x} \times 3^{y}$ , we get $x = 6$ and $y = 3$

x-y = 6-3 =3