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Let x be the smallest positive integer satisfying simultaneously x ≡…
Question
Let x be the smallest positive integer satisfying simultaneously x ≡ 2 (mod 3), x ≡ 3 (mod 5), x ≡ 2 (mod 7), x ≡ 5 (mod 11), and x ≡ 8 (mod 13). Find the sum of the decimal digits of x.
✓ Verified answer: 23checked by our engine — not a guess
Step-by-step solution
The moduli 3,5,7,11,13 are pairwise coprime with product M = 15015.
By the Chinese Remainder Theorem there is a unique solution modulo 15015.
Solving the system step by step (combine x≡2 mod3 and x≡3 mod5 → x≡8 mod15; then with x≡2 mod7 → x≡23 mod105; then with x≡5 mod11 → x≡863 mod1155; then with x≡8 mod13 → x≡6638 mod15015).
The smallest positive solution is x = 6638.
Its digit sum is 6+6+3+8 = 23.
Final answer23
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