In what number base is the addition 465 + 24 + 225 = 1050?

Analyze the question: We need a base b in which 465 + 24 + 225 = 1050, with all numbers written in base b.

Concept: A number like 465 in base b means 4b² + 6b + 5. Convert each number to an expression in b, then solve.

Solution:

• 465 = 4b² + 6b + 5
• 24 = 2b + 4
• 225 = 2b² + 2b + 5
• 1050 = b³ + 0b² + 5b + 0 = b³ + 5b

Left side: 4b² + 6b + 5 + 2b + 4 + 2b² + 2b + 5 = 6b² + 10b + 14

Set equal to the right side: b³ + 5b = 6b² + 10b + 14 b³ − 6b² − 5b − 14 = 0

Test the options:

• b = 7: 343 − 294 − 35 − 14 = 0 ✓
• b = 8: 512 − 384 − 40 − 14 = 74 ✗
• b = 9: 729 − 486 − 45 − 14 = 184 ✗
• b = 10: 1000 − 600 − 50 − 14 = 336 ✗

So b = 7.

Check in base 7 (column by column):

• Units: 5 + 4 + 5 = 14 = 2×7 + 0 → write 0, carry 2
• Sevens: 6 + 2 + 2 + carry 2 = 12 = 1×7 + 5 → write 5, carry 1
• 49s: 4 + 2 + carry 1 = 7 = 1×7 + 0 → write 0, carry 1
• Next: 1

Result: 1050 ✓

Tip: When the digits in a base question include 6, the base must be at least 7. That eliminates nothing here, but you can quickly test the options in the units column: the units sum 5 + 4 + 5 = 14 must end in 0 in the answer, so 14 must be a multiple of the base. Only 7 divides 14, so the base is 7 straight away.