Knowing that a positive integer has a remainder of 9 when divided by 10, 8 when divided by 9, and 7 when divided by 8, find the smallest positive integer that meets the conditions, and write a brief reasoning process.
已知一个正整数被10余9，被9除余8，被8除余7，求适合条件的最小正整数，写出简要的推理过程．
<gadget id="calculator">2 * 2 * 2 * 3 * 3 * 5</gadget>
<output>360</output>

<gadget id="calculator">360 - 1</gadget>
<output>359</output>

<result>359</result>