The positive integers from 1 to 150 inclusive are placed in a 10 by 15 grids so that each cell contains exactly one integer. Then the multiples of 3 are given a red mark, the multiplesof 5 are given a blue mark , and the multiples of 7 are given a green mark. How many squares have more than 1 mark?
 A 10
 B 12
 C 15
 D 18
 E 19
 

Dear Student,

The squares that receive more than one mark are those which contain integers which are divisible
by at least two of the integers 3, 5 and 7.

The integers 3 and 5 are co-prime. They have no common factor other than 1. Therefore the
integers that are divisible by 3 and 5 are those that are divisible by 3 × 5 = 15

The number of integers in the range from 1 to 15 that are divisible by 15  $= \frac{150}{15} = 10$

The numbers are $\left\{15,30,45,60,75,90,105,120,135,150\right\}$

Similarly, the integers that are divisible by 3 and 7 are those that are divisible by 3 × 7 = 21. 
So the number of positive integers in the range from 1 to 150 that is divisible by 21  
The numbers are 


Similarly, the integers that are divisible by 5 and 7 are those that are divisible by 5 × 7 = 35. 
So the number of positive integers in the range from 1 to 150 that is divisible by 35  
The numbers are $\left\{35,70,105,140\right\}$
We have 10 + 7 + 4 = 21 such numbers.

However the number 105 is in all the three lists. So to avoid over-counting we have to substract 2.
So number of such numbers = 21-2 = 19. 


Regards,