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Problems with normal subgroups

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Let N_1, ..., N_k be normal subgroups of a finite group G. If G = N_1N_2 ...N_k (the set of all elements of the form a_1a_2...a_k with a_ j in N_ j) and |G| = |N_1| . |N_2| . |N_3| ... |N_k|, prove that G = N_1 x N_2 x N_3 x ... x N_k.

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This solution helps with a problem involving normal subgroups.

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