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Number theory Difficulty 3.0 AMC 10/12 Find the answer United Kingdom

How many five-digit positive integers have the product of their digits equal to 1000?

(A) $10$ (B) $20$ (C) $28$ (D) $32$ (E) $40$

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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.