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Number theory Difficulty 2.0 Junior Find the answer United Kingdom

One of the following cubes is the smallest cube that can be written as the sum of three positive cubes. Which is it?

(A) $27$ (B) $64$ (C) $125$ (D) $216$ (E) $512$

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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.