how many positive integers between 10 and 2016 are divisible by 3 and have all the digits same
Answers
Answer:
12
Step-by-step explanation:
The required numbers are 33,66,99,111,222,333,444,555,666,777,888,999
Let's find the positive integers between 10 & 100 first, and then between 100 & 1000 and finally 1000 & 2016.
Between 10 & 100
The number should be divisible by 3, and have all the digits same.
If a two digit number has all digits same, it should be divisible by 11.
Now, We can say the numbers divisible by both 3 & 11 are required.
11 × 3 = 33 is the first one.
33 × 2 = 66
33 × 3 = 99
Between, 10 & 100 There are 3 such numbers.
Between 100 & 1000
The number should be divisible by 3, and have all the digits same.
If a three digit number has all digits same, it should be divisible by 111.
Now, We can say the numbers divisible by both 3 & 111 are required.
Firstly 111 is divisible by 3, So we will just look for numbers that are divisible by 111 ( As 111 is already divisible by 3)
111 × 1 = 111
222, 333, 444, 555, 666,777,888, 999
We get 9 such numbers between 100 & 1000.
Between 1000 & 2016
The number should be divisible by 3, and have all the digits same.
If a four digit number has all digits same, it should be divisible by 1111.
Now, We can say the numbers divisible by both 3 & 1111 are required.
1111 × 3 = 3333
3333 × 2 = 6666
3333 × 3 = 9999
Between 1000 & 2016, No such number is possible.
Finally We conclude that, There are (3+9)= 12 numbers between 10 & 2016 that are divisible by 3 and have all digits same.