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do we have to have the same amount of significant numbers in the values we got by convertion?

No. Your conversion factors may have some lack of precision.

Examples 60sec=1min infinitely precise, a definition.
1.6km= 1mile two sig figures, because the mile to km is a measured figure. 1 mile = 1.609344 kilometers would be to eight figures. So conversions depend on how precise the conversion factor is. Generally, it is wise to make the conversion factors more precise than the data being converted.

I don't know what you mean by convertion, but there may be no more significant figures in an answer than in the least significant number used to obtain the answer.

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