AMC 8 Daily Practice Round 6

Complete problem set with solutions and individual problem pages

Problem 3 Medium

What is the value of 169 - 168 - 167 + 166 + 165 - 164 - 163 + 162 + \dots + 9?

  • A.

    0

  • B.

    -9

  • C.

    5

  • D.

    9

  • E.

    84

Answer:D

By observing the arithmetic expression, we notice that the sum of the first four numbers is 0: 169 - 168 - 167 + 166 = 0

Examining the next set of four numbers, we find their sum is also 0: 165 - 164 - 163 + 162 = 0

Thus, the original expression can be rewritten as: 169 - 168 - 167 + 166 + 165 - 164 - 163 + 162 + \dots + 9 = (169 - 168 - 167 + 166) + (165 - 164 - 163 + 162) + \dots + 9

The problem now reduces to determining how many such groups exist. From 169 to 9, there are a total of 161 numbers.

When divided into groups of four, we get: 161 \div 4 = 40 groups with a remainder of 1.

Therefore, the original expression can be represented as: (169 - 168 - 167 + 166) + (165 - 164 - 163 + 162) + \dots + 9= 40 \times 0 + 9 = 9

The final result is \boxed{9}.