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Answer:
There are six integers between 197 and 303 that are divisible by 4 and 10.
Further Explanation
Number Theory is the process closely studying about integers and their patterns.
The problem deals with the integers between 197 and 303. This means that the integers 197 and 303 are excluded from the list. Furthermore, the two integers are not divisible by 4 and 10 too. By the way, when we say that an integer is divisible by another integer, this means that the remainder or residual should be zero.
1. The integer should be a multiple of 4.
2. The integer should be a multiple of 10.
Solution:
Find the LCM of 4 and 10. For any two integers, say a and b, the LCM or Least Common Multiple is the smallest possible number that is evenly divisible by both a and b.
To solve for the LCM of 4 and 10, we can employ the prime factorization process. The product of all the prime factors of the two numbers is the LCM.
[tex]\begin{aligjned}4=&\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:2\cdot{2}\\10=&\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:2\:\:\:\:\:\cdot{5}\\\\\text{Product of Prime Factors}=&2\cdot{2}\cdot{5}=20\end{aligned}[/tex]
LCM(4, 10) = 20
So from the given range of integers the first integer that is divisible by 4 and 10 immediately before 197 is 180
The second integer that is divisible by 4 and 10 immediately after 303 is 300
Take the difference of the two numbers, 300 and 180. Then divide the result by the LCM value to determine the number of integers between 197 and 303 that are divisible by 4 and 10.
[tex]\begin{aligned}300-180&=120\\\\\frac{120}{20}&=6\end{aligned}[/tex]
This means that there are six integers between 197 and 303 that are divisible by 4 and 10. These are 200, 220, 240, 260, 280, and 300.
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