To find the composite numbers between 13 and 23, we first need to identify the composite numbers in that range.
Composite numbers are defined as numbers that have more than two distinct positive divisors, meaning they can be divided by numbers other than 1 and themselves.
Let's list the integers between 13 and 23:
- 14 (composite: divisible by 1, 2, 7, 14)
- 15 (composite: divisible by 1, 3, 5, 15)
- 16 (composite: divisible by 1, 2, 4, 8, 16)
- 17 (prime: divisible only by 1 and 17)
- 18 (composite: divisible by 1, 2, 3, 6, 9, 18)
- 19 (prime: divisible only by 1 and 19)
- 20 (composite: divisible by 1, 2, 4, 5, 10, 20)
- 21 (composite: divisible by 1, 3, 7, 21)
- 22 (composite: divisible by 1, 2, 11, 22)
Now, the composite numbers between 13 and 23 are:
- 14
- 15
- 16
- 18
- 20
- 21
- 22
Now we will sum these composite numbers:
\[ 14 + 15 + 16 + 18 + 20 + 21 + 22 \]
Calculating this step by step:
- \( 14 + 15 = 29 \)
- \( 29 + 16 = 45 \)
- \( 45 + 18 = 63 \)
- \( 63 + 20 = 83 \)
- \( 83 + 21 = 104 \)
- \( 104 + 22 = 126 \)
Thus, the sum of the composite numbers between 13 and 23 is 126.
Justification:
The options presented:
A) The composite Numbers between 13 and 23 are 14 15 16 17 18 20 21 and 22 The sum of these numbers is 143 (incorrect, includes 17 which is prime)
B) The composite Numbers between 13 and 23 are 14 15 16 18 20 and 22 The sum of these numbers is 105 (incorrect, misses 21)
C) The composite numbers between 13 and 23 are 14 15 16 18 20 21 and 22. The sum of these numbers is 137 (incorrect, calculation error)
D) The composite numbers between 13 and 23 are 14 15 16 18 20 21 and 22. The sum of these numbers is 126 (correct)
Therefore, the correct answer is D: the composite numbers sum to 126.
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