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Sarah chooses two numbers $a$ and $b$ from the set $\{1, 2, 3, \ldots, 26\}$. The product $ab$ is equal to the sum of the remaining 24 numbers. What is the difference between $a$ and $b$?

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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, ordering) added by this project. Solutions are the publisher's, linked not copied.