35 *** replace asterisks with even figures so that the half-scientific number was divided into 90. how many various numbers can it will turn out? prove the answer.

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The number is divided into 90 in only case when it is divided into 9 and 10. 35 ABCs are divided into 10 only if the last figure — 0: 35ab0. On 9 share those natural numbers which sum of figures is multiple 9. That is 9 | (3 + 5 + a + b + 0) ⇔ 9 | (8 + a + b) that is possible only at (a, b) = (1, 0) = (0, 1), (1, 9), (9, 1). Answer's : 35100, 35010, 35190, 35910.
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