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d_1+d_2+...+d_6= 36 iff (d_1-6) (d_2-6) ... (s_6 -6) = -36, digits \ne 6

Source: 2007 Moldova JBMO TST p1

February 20, 2021
number theoryDigitsSumProductsum of digits

Problem Statement

The numbers d1,d2,...,d6d_1, d_2,..., d_6 are distinct digits of the decimal number system other than 66. Prove that d1+d2+...+d6=36d_1+d_2+...+d_6= 36 if and only if (d1−6)(d2−6)...(d6−6)=−36(d_1-6) (d_2-6) ... (d_6 -6) = -36.
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