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Estonian Math Competitions

Estonia algebra

Problem

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 , consisting of ones, works. Its product of digits is and digit sum .

(b) The number works. Its product of digits is and digit sum .

(c) The number has product of digits and digit sum .
Final answer
(a) Yes; (b) Yes; (c) Yes

Techniques

Integers