Understanding Divisibility by 3
A number is divisible by 3 if the sum of its individual digits is also divisible by 3. This is a fundamental rule in number theory used to quickly check divisibility without performing the actual division.
Applying the Rule to 23k57
The number given is 23k57. Here, 'k' represents a missing digit. To determine if this number is divisible by 3, we need to sum its digits:
Sum of digits = $2 + 3 + k + 5 + 7$
Calculating the sum of the known digits:
Sum = $(2 + 3 + 5 + 7) + k$
Sum = $17 + k$
Finding the Least Value for 'k'
For the number 23k57 to be divisible by 3, the sum of its digits, $17 + k$, must be a multiple of 3. We are looking for the least value of 'k'. Since 'k' is a digit, it can range from 0 to 9.
Let's test values for 'k' starting from 0:
- If $k = 0$, the sum is $17 + 0 = 17$. Is 17 divisible by 3? No ($17 \div 3 = 5$ remainder 2).
- If $k = 1$, the sum is $17 + 1 = 18$. Is 18 divisible by 3? Yes ($18 \div 3 = 6$).
Since we found a value of 'k' (which is 1) that makes the sum of the digits divisible by 3, and we started testing from the smallest possible digit (0), this value 'k = 1' is the least value required.
The number would be 23157, and the sum of its digits is $2 + 3 + 1 + 5 + 7 = 18$, which is divisible by 3.
Other possible values for k would be 4 (sum = $17 + 4 = 21$) and 7 (sum = $17 + 7 = 24$), but the question specifically asks for the least value.
Conclusion
The least value of k that makes the number 23k57 divisible by 3 is 1.
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Important Questions from Divisibility and Remainder
If the 8-digit number 888x53y4 is divisible by 72, then what is the value of (7x + 2y), for the maximum value of y?
If all positive divisors of 132 are arranged in descending order, then what digit will be at unit place of first divisor ?
If 3 2019 is divided by 10, then what is the remainder?
The number 3798125P369 is divisible by 7. What is the value of the digit P?
Consider all 3-digit numbers (without repetition of digits) obtained using three non-zero digits which are multiples of 3. Let S be their sum.
Which of the following is/are correct?
1. S is always divisible by 74.
2. S is always divisible by 9.
select the correct answer using the code given below:
