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Strong reducing agents like $\ce{LiAlH4}$ can reduce amides to amines. Sodium in ethanol (as a source of protons) is also a very strong reducing agent. Could it be therefore used to, for example, turn formamide into methylamine?

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Yes, that is used as a pathway for synthesis of amines from amides, albeit rarely.

I quote from the abstract of a communication that reports the same: (emphasis mine)

The reduction of amides to their corresponding amines has been used extensively in the synthesis of many drugs and their intermediates. This reaction has usually been carried out by means of metal hydrides (especially lithium aluminium hydride) or by catalytic hydrogenation over various catalysts. Metal and alcohol has been used for the reduction of ketones to their corresponding hydroxy compounds and oximes to their corresponding amines, but the reduction of amides under these conditions, although previously reported, is not well documented in major works of organic chemistry. This procedure is specially useful for industry since it is safer, less expensive, requires less time and provides higher yields (65-80 per cent) when compared with lithium aluminium hydride or catalytic reduction methods.

The emphasized text is written in a slightly confusing language but nonetheless confirms that amides can be reduced to amines using $\ce{Na/ROH}$, otherwise known as Bouveault-Blanc reduction.

amide to amine reduction


References:

Bhandari, K. , Sharma, V. L. and Chatterjee, S. K. (1991), ChemInform Abstract: A Convenient Method for the Reduction of Amides to Their Corresponding Amines.. ChemInform, 22: no-no. DOI: 10.1002/chin.199118077

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