Question

How many positive integers less than \[1000\] are multiple of \[5\] and are equal to \[3\] times an even integer?

Answer 1

Hint: In the given, we had asked how many positive integers less than the given number are the multiples of the given number which is equal to that number times an even integer. Here we will find out the common number which is the multiple of both the numbers and then further find out the total number of integers under the given condition.

Complete step-by-step answer:
 Here we are given that we have to find out the total number of positive integers which are less than \[1000\] which are multiple of \[5\]and are equal to \[3\] times the given even integer.
Here we are to ask that the number must be the multiple of \[5\] and that number must be the multiple of \[5\] and that number should also be equal to \[3\] times an even integer. Therefore we can say that the integer must be divisible by both\[5\] and \[3\] times an even integer. Let the smallest even integer be \[2\], therefore \[3\] times the even integer be \[6\]. So we have to find out that the number must be multiple of \[5\]and \[6\] which implies the integer must be divisible by \[5\] and \[6\]. Therefore integer must be divisible by \[5 \times 6{\text{ = }}30\]
Now, the total number of integers which are less than \[1000\] which are multiple of \[5\] are equal to \[3\] times an even integer is given by \[\dfrac{{1000}}{{30}}{\text{ }} = {\text{ }}33.33\]
Since integer cannot be in decimals. Therefore there is a total of \[33\] integers which are less than \[1000\] and also the multiple of \[5\] and are equal to \[3\] times an even integer.

Note: Integer is nothing but the numbers ranging from negative infinity to positive infinity. Zero is also included in integers. Decimals are not included in integers. Integers are also not fractional parts or rational numbers. This is the reason that we have taken\[33\] from \[33.33\] numbers because \[0.33\] is not counted as integers and \[33\] are complete integers.