1. **State the problem:** We have a multiplication problem where 407 is multiplied by a two-digit number 6☐ (where ☐ is the missing digit). The partial products and the final sum are given in base 5 and base 4 respectively, and we need to find the missing digit. 2. **Analyze the given information:** - The first partial product is 3256 in base 5. - The second partial product is 24420 in base 4. - The final sum is 27676 (no base specified, assume base 10 for the sum). 3. **Convert the partial products to base 10:** - Convert 3256 base 5 to base 10: $$3 \times 5^3 + 2 \times 5^2 + 5 \times 5^1 + 6 \times 5^0$$ Note: Digit 6 is invalid in base 5 (digits must be 0-4), so this suggests a typo or the base is not base 5. Since 6 is present, the base must be at least 7. But the problem states 3256₅, so 6 is invalid. Let's assume the subscript is a typo and treat it as base 7. - Convert 3256 base 7 to base 10: $$3 \times 7^3 + 2 \times 7^2 + 5 \times 7^1 + 6 \times 7^0 = 3 \times 343 + 2 \times 49 + 5 \times 7 + 6 = 1029 + 98 + 35 + 6 = 1168$$ - Convert 24420 base 4 to base 10: $$2 \times 4^4 + 4 \times 4^3 + 4 \times 4^2 + 2 \times 4^1 + 0 \times 4^0 = 2 \times 256 + 4 \times 64 + 4 \times 16 + 2 \times 4 + 0 = 512 + 256 + 64 + 8 + 0 = 840$$ 4. **Set up the multiplication:** - Let the missing digit be $x$. - The multiplier is $6x$ in base 10. - The multiplicand is 407. 5. **Relate partial products to digits:** - The first partial product corresponds to $407 \times 6 = 2442$ in base 10, but we have 1168 from above, so the base assumption might be incorrect. 6. **Re-examine the problem:** - Since the partial products are given in different bases, the problem likely involves base arithmetic. - The first partial product 3256₅ is invalid due to digit 6. 7. **Assuming the subscripts are the bases of the partial products:** - 3256₅ is invalid, so the base must be at least 7. - 24420₄ is invalid because digit 4 is not allowed in base 4. 8. **Conclusion:** The problem as stated contains inconsistent base notation or typos. **Therefore, the missing digit cannot be determined with the given information.**