Divisibility Rules for 2012
Divisibility rules are powerful tools used in mathematics to determine whether a number is divisible by another number without performing the division itself. One such interesting example is the divisibility rule for 2012. This article will explore the specific rule for 2012 and related rules for 2013, 2014, and other composite numbers. We will also examine how these rules can be generalized to other numbers.
Divisibility Rule for 2012
Let's begin with the divisibility rule for 2012. Since 2012 can be factored into 4 x 503, a number N is divisible by 2012 if and only if it is both divisible by 4 and 503. Therefore, we can break down the problem into two parts. This section will focus on the process of determining the divisibility of a number by 503, as the divisibility by 4 can be handled directly.
Step-by-Step Process for 503
Suppose you want to check whether a number N is divisible by 2012. Here’s a step-by-step method to determine if N is divisible by 503:
Check if N is divisible by 4. If not, then N is not divisible by 2012. If it is, divide N by 4 to simplify N. If N is even, replace N by N/2. If N is odd, replace N by N - 503/2. Continue this process until N equals 503. At this stage, the divisibility of N by 503 is obvious. To further optimize the process, you can replace step 2 with the following: 2' If N is odd, replace N by N - 50310^n for a large enough n. This step works because if N is divisible by 503, then so is N - 50310^n.Let's illustrate with an example. Consider N 179963340. Since 340 is divisible by 4, N is also divisible by 4. Simplifying, we get N 179963340 / 4 44990835. Following the steps:
44990835 → 22495166 → 11247583 → 5623540 → 2811770 → 1405885 → 702691 → 351094 → 175547 → 87522 - 43761 → 21629 → 10563 → 5030 Since 5030 is divisible by 503, the original number N is divisible by 503. By extension, N is also divisible by 2012.Combining Rules for 2 and 503
To determine if a number N is divisible by 2012, it must satisfy two conditions:
Divisibility by 2: The last digit of the number must be even. Divisibility by 503: Multiply the last digit by 151, add the truncated digits, and check if the result is divisible by 503. Repeat as necessary.Let's apply this to the number 179963340 again:
The last digit is 0, which is even, so the number is divisible by 2. Using the 503 rule, multiply the last digit by 151 and add the truncated digits: 0 x 151 17996334. This simplifies to 17996334. Checking divisibility, we get 17996334 / 503 35746, which is an integer, so the number is divisible by 503.Therefore, the number 179963340 is divisible by 2012.
Generalization of Divisibility Rules
Divisibility rules can be generalized to other numbers. For instance, consider a number y represented as dn-1 dn-2 ——— dn-1 dn-2 d3 d2 d1 d0 where d represents the digits. To test if this number is divisible by 12, sum all the digits from dn-1 to d3 and multiply by 4. Add this to the value formed by the digits d1d0. If the result is divisible by 12, then y is also divisible by 12.
Examples of Generalized Divisibility Rule
Example 1: Divisibility of 144 by 12
For the number 144, sum the digits from 1 to 4 and multiply by 4. Then add this to the last two digits 44:
Sum 1 x 4 44 48 48 is divisible by 12 (12 x 4 48)Therefore, 144 is divisible by 12.
Example 2: Divisibility of 46134 by 12
For the number 46134, sum the digits from 4 to 1 and multiply by 4. Then add this to the last two digits 34:
Sum 461 x 4 34 4434 34 4468 4468 is not divisible by 12 (12 x 372 4464)Therefore, 46134 is not divisible by 12.
Divisibility by 12
A number is divisible by 12 if and only if it is divisible by both 3 and 4. Here are the specific rules:
Divisibility by 3: Check if the sum of the digits of the number is divisible by 3. Divisibility by 4: Check if the last two digits of the number are divisible by 4.Conclusion
The divisibility rules for 2012 provide a clear and concise method to determine divisibility, which can be utilized in a variety of mathematical contexts. By understanding and applying these rules, one can efficiently test divisibility without resorting to direct division. These rules are also useful in verifying the correctness of arithmetic operations and in solving mathematical problems involving large numbers.