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Hey oomis – good question. The Jewish calendar is comprised of 12 lunar months of 29.5 days each. That’s a 354 day year. Since we need to align with the solar calendar (so that Pesach is not in the spring, as you said) we need a leap month. We need one rather frequently (7 times in every 19 years) because each year we fall more than 11 days behind.
The same is true for the non-Jewish calendar, though to a lesser degree. Each year we are off by about 1/4 of a day. That doesn’t make a noticable difference, but after a while it builds up. Every 100 years we would have our seasons off by nearly a month! Imagine spring not coming until late April, summer starting in late July, etc. After 400 years, we’d be off by more than 3 months, so summer would start in September, and winter in March, etc. Not quite as dramatic, but still problematic. You’d hear of the battle of Leningrad and wonder why it got colder in January than when the siege started in September.