Identifying the Hidden Pattern

The problem presents an equation, "137 + 276 = 435", which is incorrect under standard decimal (base-10) arithmetic. This indicates that the numbers are represented in a different number base.

Step 1: Determine the Number Base

We need to find a base 'b' where the equation $137_b + 276_b = 435_b$ is true. Let's test base 8 (since all digits used are less than 8).

First, we convert the numbers from base 8 to base 10:

• $(137)_8 = 1 \times 8^2 + 3 \times 8^1 + 7 \times 8^0 = 64 + 24 + 7 = 95_{10}$
• $(276)_8 = 2 \times 8^2 + 7 \times 8^1 + 6 \times 8^0 = 128 + 56 + 6 = 190_{10}$

Now, we add the base-10 values:

$95 + 190 = 285_{10}$

Finally, we check if $(435)_8$ is equal to $285_{10}$:

$(435)_8 = 4 \times 8^2 + 3 \times 8^1 + 5 \times 8^0 = 256 + 24 + 5 = 285_{10}$

Since the values match, the numbers are in base 8.

Step 2: Calculate the Second Sum in Base 8

Now we must calculate $731 + 672$ in base 8. We can do this by converting to base 10, adding, and then converting back to base 8.

Convert to base 10:

• $(731)_8 = 7 \times 8^2 + 3 \times 8^1 + 1 \times 8^0 = 448 + 24 + 1 = 473_{10}$
• $(672)_8 = 6 \times 8^2 + 7 \times 8^1 + 2 \times 8^0 = 384 + 56 + 2 = 442_{10}$

Add in base 10:

$473 + 442 = 915_{10}$

Convert the result back to base 8:

• 915 ÷ 8 = 114 remainder 3
• 114 ÷ 8 = 14 remainder 2
• 14 ÷ 8 = 1 remainder 6
• 1 ÷ 8 = 0 remainder 1

Reading the remainders from bottom to top, we get $(1623)_8$.

Conclusion

The result of the sum is 1623.