The sum of all the elements of the set 1,2,,100 : (,24)=1 is ___ Answer: 1633

Question

The sum of all the elements of the set $\{\alpha \in \{1,2,\ldots,100\} : \gcd(\alpha,24)=1\}$ is \_\_\_

Answer: 1633

Accepted Answer

The numbers up to 24 coprime to 24 are: $1,5,7,11,13,17,19,23$ --- sum $=96$.

This pattern repeats every block of 24 (with each block's coprime numbers summing to $96+24\times8\times(\text{block index})$ adjustment). Over four complete blocks (1--24, 25--48, 49--72, 73--96) plus the partial block (97--100, contributing just 97):

$$\text{Sum} = 96 + (24\times8+96) + (48\times8+96) + (72\times8+96) + 97 = 1633$$

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