GG.19 Three digit integers divided by 35 and 40

When 100 is divided by 35, a quotient is 2 and remainder is 30. 

When 100 is divided by 40, a quotient is 2 and remainder is 20. 

Find the number of three digits integer including 100 which a quotient becomes the same, when it is divided by 35 and 40. 

And also find the largest integer among them.

Answer

55 pieces
244

Solution

According to that 35 × 2 = 70 and 35 × 3 = 105, the range of the integer from which a quotient is 2 if it is divided by 35 is 70 ≦ X ≦ 104.
According to that 40 × 2 = 80 and 40 × 3 = 120, the range of the integer from which a quotient is 2 if it is divided by 40 is 80 ≦ X ≦ 119. 

Thus, the three digit integers from which a quotient is 2 are five pieces in 100 or more and 104 or less.
Arranges the range of the three digit integers of which quotient becomes 3, 4, 5, ....., it will become as it is shown in a lower table. 

It is 20 pieces among 120 or more and 139 or less that a quotient is 3.
It is 15 pieces among 160 or more and 174 or less that a quotient is 4
It is 10 pieces among 200 or more and 209 or less that a quotient is 5
It is 5 pieces among 240 or more and 244 or less that a quotient is 6.
There is no integer overlapped that a quotient is 7.
Therefore total number of integer is 5 + 20 + 15 + 10 + 5 = 55 pieces.
And it comes out among these and the biggest integer is 244.