How many positive integers less than 1000 have the property that the sum of the digits of each such number is divisible by 7 and the number itself is divisible by 3
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Lets denote sum of the digits if a positive integer as
For a number to be divisible by 3, the should be divisible by 3
Also, we want integers whose is divisible by 7
Therefore we want positive integers whose is divisible by 21
Now maximum for a 1-digit and 2-digit number is 9 and 18 respectively.
And since we want integers less than 1000, we want to find integers such as , i.e., 3-digit numbers only
The pair of 3 digits which has sum of 21 are
1 of the above 8 pair has 3 same digits and the number of such integers can be given as
3 of the above 8 pairs has 2 same digits and the number of such integers can be given as
4 of the above 8 pairs has no common digits and the number of such integers can be given as
Positive integers with divisible by 3 and 7
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For a number to be divisible by 3, the should be divisible by 3
Also, we want integers whose is divisible by 7
Therefore we want positive integers whose is divisible by 21
Now maximum for a 1-digit and 2-digit number is 9 and 18 respectively.
And since we want integers less than 1000, we want to find integers such as , i.e., 3-digit numbers only
The pair of 3 digits which has sum of 21 are
1 of the above 8 pair has 3 same digits and the number of such integers can be given as
3 of the above 8 pairs has 2 same digits and the number of such integers can be given as
4 of the above 8 pairs has no common digits and the number of such integers can be given as
Positive integers with divisible by 3 and 7
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