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What is the sum of all the integers from −7 to 7, taking both ends and everything in between?

Class 7Integers (Extended)Integers
  • A49
  • B−7
  • C0
  • D7

Step-by-Step Solution

Pair every number with its opposite: , and so on down to , leaving only the 0 in the middle. Every pair is an additive-inverse pair, so the whole sum is 0. Answering 49 multiplies instead of adding, and −7 comes from imagining one leftover negative that is not there.
Answer: C) 0

Key idea: Properties — same as for whole numbers, mostly

Integers obey:
- **Commutative** for addition and multiplication: a + b = b + a; a × b = b × a.
- **Associative** for addition and multiplication: (a + b) + c = a + (b + c); (a × b) × c = a × (b × c).
- **Distributive**: a × (b + c) = a × b + a × c — and this is true even when a, b, c are negative.

Subtraction and division are NOT commutative or associative for integers (they aren't for whole numbers either). Order and grouping matter.

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