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Number theory Difficulty 2.0 Junior Prove it United Kingdom

A two-digit number is divided by the sum of its digits. The result is a number between 2.6 and 2.7. Find all of the possible values of the original two-digit number.

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Source: UK Mathematics Trust, licensed © UK Mathematics Trust; past papers published free at ukmt.org.uk. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.