For a transformer output (secondary) of 6 volts, you would need a transformer input (primary) of 1 1/2 volts, because the question tells you that secondary = 4 times primary.
So you 'could' use a 1 1/2 volt battery, but it's could, not would, because as Beso correctly explains, batteries provide DC (direct current) and transformers only work on AC (alternating current) both in and out.
In principle (but not very realistically) you can imagine an alternator of some kind converting the 1 1/2 volt DC battery input to 1 1/2 volts AC, then the transformer stepping that up to 6 volts AC, then a rectifier converting that to the 6 volts DC that the phone needs. It's possible.