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  "body": "\n## Statement\n\nLet $p$ be a prime number, and let $m$ and $n$ be natural numbers.\nThen $\\nu_p \\binom{m + n}{m}$ is the number of carries when one adds $m$ and $n$ in base $p$. Here $\\nu_p$ is the [[p-adic valuation|04VCPS30ZE]].\n\n## Remarks\n\nFor example, in base $2$, $3 + 1 = 11_2 + 1_2 = 100_2$ has $2$ carries, and $\\binom{4}{1} = 4 = 2^2$.\nThe theorem follows from [[Legendre's formula|01WPV9PHN7]] in the form with the digit sums $s_p$.\nA consequence: $p \\mid \\binom{m+n}{m}$ if and only if there is at least one carry.\n\n## History\n\nErnst Kummer proved the theorem in 1852.\n\n## Explanations\n\n### L1 human\n\nWrite $m$ and $n$ in base $p$ and add them by hand. Count how many times you carry a $1$.\nThat count is the exponent of $p$ in the binomial coefficient $\\binom{m+n}{m}$.\nFor $p = 10$ the rule fails, because $10$ is not prime. The theorem needs a prime base.\n",
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