<p>For <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> a positive integer, let <img alt="$R(n)$" height="20" src="https://latex.artofproblemsolving.com/a/e/9/ae95f1464015b0c1827d95ac028fd6c38b34e401.png" width="41" /> be the sum of the remainders when <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> is divided by <img alt="$2$" height="12" src="https://latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png" width="8" />, <img alt="$3$" height="12" src="https://latex.artofproblemsolving.com/7/c/d/7cde695f2e4542fd01f860a89189f47a27143b66.png" width="8" />, <img alt="$4$" height="12" src="https://latex.artofproblemsolving.com/c/7/c/c7cab1a05e1e0c1d51a6a219d96577a16b7abf9d.png" width="9" />, <img alt="$5$" height="12" src="https://latex.artofproblemsolving.com/7/9/0/79069377f91364c2f87a64e5f9f562a091c8a6c1.png" width="8" />, <img alt="$6$" height="12" src="https://latex.artofproblemsolving.com/6/0/1/601a7806cbfad68196c43a4665871f8c3186802e.png" width="8" />, <img alt="$7$" height="12" src="https://latex.artofproblemsolving.com/e/0/a/e0a0db32027a732ac57d37ef2ae9bb150f65b108.png" width="9" />, <img alt="$8$" height="14" src="https://latex.artofproblemsolving.com/8/4/5/8455f3b5cb3b4880b8c9d782a5c1f0334db819eb.png" width="11" />, <img alt="$9$" height="12" src="https://latex.artofproblemsolving.com/b/f/2/bf2c9074b396e3af0dea52d792660eea1c77f10f.png" width="8" />, and <img alt="$10$" height="13" src="https://latex.artofproblemsolving.com/f/c/6/fc606f7f1e530731ab4f1cc364c01dc64a4455ee.png" width="17" />. For example, <img alt="$R(15) = 1+0+3+0+3+1+7+6+5=26$" height="20" src="https://latex.artofproblemsolving.com/5/b/7/5b79cf4916be912c5d86bc49376e62d16df40db1.png" width="374" />. How many two-digit positive integers <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> satisfy <img alt="$R(n) = R(n+1)\,?$" height="20" src="https://latex.artofproblemsolving.com/a/d/1/ad11bc745c9ea1a17fe843b746debf4423cea307.png" width="147" /></p>

Question No: 433

For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$ , $3$ , $4$ , $5$ , $6$ , $7$ , $8$ , $9$ , and $10$ . For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$ . How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$

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Tags: Number Theory

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