I'm sure Luke Strawwalker will chime-in on this, but the short version is that Saturns II, III and IV were designs that didn't make the cut as the lunar mission evolved. I'm pretty sure the Saturn V was the Saturn IV with an extra engine so the "number of engines" reason is not that far off.

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NAR 79743
NARTrek Silver
I miss being SAM 062

Awaiting First Launch: Too numerous to count
Finishing: Zooch Saturn V; Alway/Nau BioArcas; Estes Expedition; TLP Standard
Repair/Rescue: Cherokee-D (2); Centuri Nike-Smoke; MX-774
On the Bench: 2650;
Dream Stage: 1/39.37 R-7