On multiplying a number by 7, the product is a number consisting only of the digit 3. Find the smallest such product.

Given data

• We need the smallest repdigit of 3s (say with k digits) that is divisible by 7.

Concept / Approach

• Let a_k be the k-digit number of all 3s. Recurrence (mod 7): a_k+1 ≡ 10·a_k + 3 (mod 7) and since 10 ≡ 3 (mod 7), a_k+1 ≡ 3(a_k + 1) (mod 7).

Step-by-step residues (mod 7)

a₁ = 3 ≡ 3a₂ ≡ 3(3 + 1) = 12 ≡ 5a₃ ≡ 3(5 + 1) = 18 ≡ 4a₄ ≡ 3(4 + 1) = 15 ≡ 1a₅ ≡ 3(1 + 1) = 6 ≡ 6a₆ ≡ 3(6 + 1) = 21 ≡ 0 ⇒ divisible by 7

Conclusion

The smallest all-3s multiple of 7 has 6 digits: 333333.

Final Answer

Smallest such product = 333333.