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Are there any sources that say it's a mitzvah to exercise- i.e., to walk around the block to keep up your health?

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Yes. G-d Commands us to exercise. The sage Chafetz Chaim wrote that “The entire Torah is dependent upon the mitzvah of taking care of your body.” The physician, Rambam, stressed the importance of proper sleep and exercise.[1] Maimonides explained:

“One should not neglect physical exercise of the body, as do people of learning who diligently study the entire day and night...” and, “one is obligated [by G-d] to refrain from things that impair the body and to adapt those elements which strengthen it.” Also, “Elderly people [in particular] require their bodies to move because their bodies require warmth. No older person should rest without having done some exercise.”

The sage Hillel said that bathing is a mitzvah [a divine command]. During the Black Plague of 1347-1351, which killed 200 million people in Europe, the Jews were spared because Jews washed their hands as pre scribed in the Bible.

[1] See his medical writings.

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    Where does the sage Chafetz Chaim say this? Where in his medical writings does the physician Rambam discuss? Where is that quote from? Is it a translation? If so, whose?
    – Alex
    Mar 25 at 3:27
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    Where does the sage Hillel say that bathing is a mitzvah? How do you know that Jews were spared from the plague? And if they were, how do you know it was because they washed their hands? And where is this prescribed in the Bible?
    – Alex
    Mar 25 at 10:31

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