Forming 5-Digit Numbers Divisible by 10 Using Digits 0, 1, 3, 5, 7, and 9
When it comes to creating 5-digit numbers that are divisible by 10 using the digits 0, 1, 3, 5, 7, and 9 without repetition, understanding the divisibility rule and permutation principles is essential. This article explores the steps and calculations needed to determine the total number of such 5-digit numbers.
Divisibility by 10
A number is divisible by 10 if and only if its last digit is 0. This is a crucial condition for this problem, as it helps us fix one of the digits first.
Steps to Solve
The process to find the solution involves several key steps, as follows:
Fix the Last Digit
Since the number must be divisible by 10, the last digit must be 0. This simplifies our task as we only need to arrange the remaining digits in the first four positions.
Choosing the First Digit
The first digit cannot be 0. We have five choices left: 1, 3, 5, 7, and 9. This gives us 5 options for the first digit.
Choosing the Remaining Digits
After selecting the first digit, we have 4 digits left (since we can't repeat digits and 0 is already used for the last position). We need to choose 3 digits from these 4 digits to fill the second, third, and fourth positions.
Arranging the Remaining Digits
The number of ways to arrange 3 digits from 4 is given by the factorial of 3, which is (3! 3 times 2 times 1 6).
Putting it all together, we calculate the total number of arrangements as follows:
Choices for the first digit: 5 (1, 3, 5, 7, 9)
Arrangements of the remaining 3 digits: (4! 4 times 3 times 2 times 1 24)
Therefore, the total number of arrangements is (5 times 24 120).
Conclusion
Thus, the total number of 5-digit numbers that can be formed using the digits 0, 1, 3, 5, 7, and 9 without repetition and which are divisible by 10 is 120.
Brute Force Approach Using J Programming Language
To validate our solution, we can use a brute force approach with the J programming language as demonstrated below.
Calculations in J
The J programming language can be used to list all possible 5-digit numbers generated from the digits 0, 1, 3, 5, 7, and 9 without repetition and check their divisibility by 10. Here is the calculation:
m . ev n~1e4 n . 10.5 perm 6{0 1 3 5 7 9 120Running the above code in J confirms that the total number of such 5-digit numbers is 120.
List of Numbers
The resulting list of these 120 five-digit numbers, all of which are divisible by 10, can be generated as follows:
m 13570 13590 13750 13790 13950 13970 15370 15390 15730 15790 15930 15970 17350 17390 17530 17590 17930 17950 19350 19370 19530 19570 19730 19750 31570 31590 31750 31790 31950 31970 35170 35190 35710 35790 35910 35970 37150 37190 37510 37590 37910 37950 39150 39170 39510 39570 39710 39750 51370 51390 51730 51790 51930 51970 53170 53190 53710 53790 53910 53970 57130 57190 57310 57390 57910 57930 59130 59170 59310 59370 59710 59730 71350 71390 71530 71590 71930 71950 73150 73190 73510 73590 73910 73950 75130 75190 75310 75390 75910 75930 79130 79150 79310 79350 79510 79530 91350 91370 91530 91570 91730 91750 93150 93170 93510 93570 93710 93750 95130 95170 95310 95370 95710 95730 97130 97150 97310 97350 97510 97530The listing confirms that there are indeed 120 such 5-digit numbers.
Additional Resources
To further explore this topic, you may want to read more about divisibility rules, permutations, and combinatorial mathematics. Oftentimes, understanding the underlying mathematical principles can help in solving more complex problems involving numbers and sequences.