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Estonian Math Competitions

Algebra Difficulty 4.9 AIME Prove it Estonia

Does there exist a positive integer whose

a. digit sum is 100 more than the product of digits?

b. product of digits is 100 more than the digit sum?

c. product of digits is 100 times greater than the digit sum?

Solution

Answer: (a) Yes; (b) Yes; (c) Yes.

(a) The number $111\ldots1$ , consisting of $101$ ones, works. Its product of digits is $1$ and digit sum $101$ .

(b) The number $1111112345$ works. Its product of digits is $120$ and digit sum $20$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.