To start off, write the ideal gas law:
PV = nRT
It is helpful to identify your constants. R is always constant (it's the ideal gas constant) and we are told that T is constant as well, at 24 degrees Celcius, or 297.15 K.
We also know that the number of moles is constant. If we're dealing with a closed container (which we can assume we are), then no moles are added or subtracted during the process.
If we are solving for the final pressure, we want to rewrite the ideal gas law:
P = nRT/V
Right off the bat we know that T = 297.15 K. The final volume is 1.90*10^2 mL; converting that to liters (so that the units cancel in the final calculation) we get 0.190 L. R, the ideal gas constant, is 0.0821 L*atm / (mol*K).
The only piece of information that we're missing is the number of moles. Recall that earlier we said the number of moles is constant, so we can use the initial conditions of the gas to determine the number of moles.
Rearrange the ideal gas law:
n = PV / RT
P = 1.68 atm
T = 297.15 K
V = 0.456 L
R = 0.0821 L*atm / (mol*K)
Plug those values in, make sure your units cancel, and you'll get the number of moles.
Remember the equation for the final pressure:
P = nRT / V
You can plug in the final volume, the temperature, the ideal gas constant R, and your calculated number of moles, to find the pressure needed to compress the gas.