A number theory problem by Madhavarapu Revanth
Number Theory
Level
2
The first digit from the left of a 4 digit number is equal to the number of zeros in the number.The second digit is equal to the number of digits 1,the third digit is equal to the number of digits 2 and the fourth digit is equal to the number of digits 3.How many numbers have this property?
1
2
0
4
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1 solution
I did this problem in this way. I let the number be ABCD.
According to the question,
A = No. of '0's
B = No. of '1's
C = No. of '2's
D = No. of '3's
Since it is a 4 digit number, values of A,B,C,D range between 0 and 3. Also, A cant be 0 cause if its 0 then the number of zeroes in the number becomes 1, which contradicts the definition of A. So, A = 1,2,3. We can figure out the no. of possible numbers by using trial and error method.
When A = 1, B will have to be 2. We get 2 options now - 1201 and 1210. 1201 cannot be an answer because C = 0 but there is one '2'. 1210 follows the rules hence it is an answer.
When A = 2, possible numbers are 200D, 20C0, 2B00. Since there is a 2 in each number C will have to be 2, so 2020 is the only possible answer here.
When A = 3, 3000 can be the only possible number. Clearly, 3000 does not follow the rules, so it is not answer.
So, the possible numbers are 1210 and 2020. Thus the answer is 2. Thank You