Question

How many positive integers that are less than 741 are not multiples of both 4 and 7?

Answer 1

`\displaystyle \\ n \ \textless \ 741 \\ n \in \{0;~ 1; ~2;~ \cdots~ ; 740\} ~~ \Longrightarrow~~ \boxed{741 \text{ numbers}} \\ \\ n \in \{0; ~1;~ 2;~ \cdots~ ; 740\} \text{ and } n~ \vdots ~4 \\ \Longrightarrow~~ n \in \{\underline{0};~4;~8;~12;~16;~20;~24;~\underline{28};~32;~ \cdots~ 740\} \\ \\ \Longrightarrow~~ (740 - 0)/(4)+1 = 185+1=\boxed{186 \text{ numbers multiples of }4}`



`\displaystyle \\ n \in \{0; ~1;~ 2;~ \cdots~ ; 740\} \text{ and } n~ \vdots ~7 \\ \Longrightarrow~~ n \in \{\underline{0};~7;~14;~21;~\underline{28};~35;~ \cdots~ 735\} \\ \\ \Longrightarrow~~ (735 - 0)/(7)+1 = 105+1=\boxed{106 \text{ numbers multiples of }7}`



`\displaystyle \\ n \in \{0; ~1;~ 2;~ \cdots~ ; 740\} \text{ and n = common multiple of 4 and 7} \\ \Longrightarrow~~ n \in \{\underline{0};~\underline{28};~\underline{56};~\underline{84}~ \cdots~\underline{728}\} \\ \\ \Longrightarrow~(728 - 0)/(28)+1 = 26+1=\boxed{27 ~\text{numbers common multiple of 4 and 7}} \\ \\ \text{How many positive integers that are } \\ \text{less than 741 are not multiples of both 4 and 7 ?} \\ \\ Answer: ~~ 740 - 186 -106 + 27 = \boxed{\boxed{475 \text{ numbers} }}`




Answer 2

439.
multiples of 7 = 117 nos.
multiples of 4 = 185 nos.
185 + 117 = 302
741 - 302 = 439