When 1 and 1000 are such integers as it is divided by 3 leaving a remainder of one.
Out of the integers from 1 to 1000, answer the following question about the integer which remainder is one when it is divide by 3.
(1) Find the number of multiple of 2 and the number of multiple of 5, respectively in the set of of the numbers being divided by 3 leaving a reminder of 1.
(2) Find the number of integers which are not the multiple of 2 or the multiple of 5, either in the set?
Out of the integers from 1 to 1000, answer the following question about the integer which remainder is one when it is divide by 3.
(1) Find the number of multiple of 2 and the number of multiple of 5, respectively in the set of of the numbers being divided by 3 leaving a reminder of 1.
(2) Find the number of integers which are not the multiple of 2 or the multiple of 5, either in the set?
Answer
(2) As shown in the Venn diagram below, B is the number of the integer which is not the multiple of 5 nor 2, which is the number subtracted the number of multiples of 2 or 5 from 334 of the number divided by 3 leaving a remainder of 1.
(1) 167 pieces, 67 pieces
(2) 134 pieces
(2) 134 pieces
Solution
(1) The integer which the remainder is one when it is divided by 3 is 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, ......, 1000.
The number is 333 + 1 = 334 pieces according to 1000 / 3 = 333 remainder 1.
Looking for the multiple of two among 334 pieces, there are 4, 10, 16, 22, 28, 34, 40 -------.
Since it turns out that that the number of multiple of two is a half of 334, it is 334 / 2 = 167 pieces.
Next, looking for the multiple of five among 344 pieces, there are 10, 25, 40, -----.
The multiple of five can be expressed as 10 + 15 × □ .
Next, looking for the multiple of five among 344 pieces, there are 10, 25, 40, -----.
The multiple of five can be expressed as 10 + 15 × □ .
It is referred to as 10 + 15 × □ = 1000 in order to find largest □.
□ = (1000 - 10) / 15 = 66
Thus, □ is from 0 to 66.
Therefore, the number of the multiple of five is 66 +1 = 67 pieces.
(2) As shown in the Venn diagram below, B is the number of the integer which is not the multiple of 5 nor 2, which is the number subtracted the number of multiples of 2 or 5 from 334 of the number divided by 3 leaving a remainder of 1.
The number of common multiple of 2 and 5 which is A in the Venn diagram is to be found.
In the set of the numbers being divided by 3 leaving a reminder of 1, the least common multiple of 2 and 5 is 10.
In the set of the numbers being divided by 3 leaving a reminder of 1, the least common multiple of 2 and 5 is 10.
The number following 10 number is a number which 30, the least common multiple of 2, 3, and 5 is added and it is 10 + 30 = 40.
The number can be expressed as 10 + 30 × □. 10 + 30 × □ = 1000, then □ = (1000 - 10) / 30 = 33.
Since □ is from zero to 33, there are 33 +1 = 34 common multiples of 2 and 5.
Therefore, the number of the integer to find is 344 - (167 + 67 - 34) =134 pieces.
