can you tell those numbers which are lying between 200 and 500 and divisible by 3 if digits are repeated?

1. First let's find all three digit numbers starting with '2' which are divisible by '3' when digits are repeated.

    2 _ _ .

Since digits have to be repeated, we have two possibilities: 2 _ 2 or 2 X X or 2 2 X, where X is any digit from '0' to '9'.

For 2 _ 2, for the number to be divisible by '3', the middle digit can be 2 or 5 or 8.

We get the numbers 2 2 2 , 2 5 8 , 2 8 2. Three numbers divisible by '3'.

For 2 X X, for the numbers to be divisible by '3', 'X' can be 2 or 5 or 8. But for X=2, we get the number 2 2 2, which has already been counted above. Hence, excluding that we get 2 5 5 and 2 8 8. Two numbers divisible by '3'.

For 2 2 X, for the numbers to be divisible by '3', 'X' can be 2 or 5 or 8. But for X=2, we get the number 2 2 2, which has already been counted above. Hence, excluding that we get 2 2 5 and 2 2 8. Two numbers divisible by '3'.

Hence, in total we get 3+2+2 =7 such numbers starting with '2' which are divisible by three when digits are repeated..... (1)

 

2. Next, let's find all three digit numbers starting with '3' which are divisible by '3' when digits are repeated.

   3 _ _ .

Since the digits have to be repeated, we have two possibilities: 3 _ 3 or 3 X X or  3 3 X, where X is any digit from '0' to '9'.

For 3 _ 3, for the number to be divisible by '3', the middle digit can be 0 or 3 or 6 or 9.

We get the numbers 3 0 3 , 3 3 3 , 3 6 3, 3 9 3. Four numbers divisible by '3'.

For 3 X X, for the numbers to be divisible by '3', 'X' can be 0 or 3 or 6 or 9. But for X=3, we get the number 3 3 3, which has already been counted above. Hence, excluding that we get 3 0 0, 3 6 6, 3 9 9. Three numbers divisible by '3'.

For 3 3 X, for the numbers to be divisible by '3', 'X' can be 0 or 3 or 6 or 9. But for X=3, we get the number 3 3 3 , which has already been counted above. Hence, excluding that we get 3 3 0 or 3 3 6 and 3 3 9. Three numbers divisible by '3'.

Hence, in total we get 4+3+3 =10 such numbers starting with '3' which are divisible by three when digits are repeated..... (2)

 

3. 

Next, let's find all three digit numbers starting with '4' which are divisible by '3' when digits are repeated.

   4 _ _ .

Since the digits have to be repeated, we have two possibilities: 4 _ 4 or 4 X X or 4 4 X, where X is any digit from '0' to '9'.

For 4 _ 4, for the number to be divisible by '3', the middle digit can be 1 or 4 or 7.

We get the numbers 4 1 4, 4 4 4, 4 7 4. Three numbers divisible by '3'.

For 4 X X, for the numbers to be divisible by '3', 'X' can be 1 or 4 or 7. But for X=4, we get the number 4 4 4, which has already been counted above. Hence, excluding that we get 4 1 1 and 4 7 7. Two numbers divisible by '3'.

For 4 4 X, for the numbers to be divisible by '3', 'X' can be 1 or 4 or 7. But for X=4, we get the number 4 4 4 , which has already been counted above. Hence, excluding that we get 4 4 1 and 4 4 7. Two numbers divisible by '3'.

Hence, in total we get 3+2+2 =7 such numbers starting with '4' which are divisible by three when digits are repeated..... (3)

 

From (1), (2) and (3), we get total number of such numbers lying between 200 and 500 as 7+10+7=24.