Answer
Using the hint, we consider pushing a child on a swing.
When you push at the natural frequency, every push is applied when the swing is moving away from you. This is effective at making the swing go higher.
However, pushing at twice the swing's frequency, i.e., twice as often, is ineffective at increasing the amplitude of the swing. Picture it: you'd be pushing opposite to the child’s velocity with every other push. The same argument can be used with higher whole number multiples of its natural frequency.
On the other hand, pushing at a submultiple of its natural frequency might work. For example, if you pushed the child at half the swing's natural frequency, you would push only every other cycle. You'd "miss some opportunities" to push, but overall, your pushes would be in sync with the child’s velocity, and the swing would move with increased amplitude.
In conclusion, a system won't resonate with multiples of its natural frequency, but will resonate with a driving force that vibrates at a submultiple of its natural frequency.
