Consider a 6-digit number of the form XYXYXY. The number is divisible by :
3, 7, 13 and 37
Analyzing the Structure of the 6-Digit Number XYXYXY
We are given a 6-digit number with a specific repeating pattern, XYXYXY. Let's break down this number based on place values to understand its structure mathematically. A number like XYXYXY can be written as:
\(\text{XYXYXY} = X \times 100000 + Y \times 10000 + X \times 1000 + Y \times 100 + X \times 10 + Y \times 1\)
Factoring the Expression for XYXYXY
Now, let's group the terms involving X and the terms involving Y:
\(\text{XYXYXY} = (X \times 100000 + X \times 1000 + X \times 10) + (Y \times 10000 + Y \times 100 + Y \times 1)\)
Factor out X from the first group and Y from the second group:
\(\text{XYXYXY} = X(100000 + 1000 + 10) + Y(10000 + 100 + 1)\)
Simplify the terms in the parentheses:
\(\text{XYXYXY} = X(101010) + Y(10101)\)
Now, we can see that 10101 is a common factor in both terms. Let's factor out 10101:
\(\text{XYXYXY} = 10101(10X + Y)\)
The number XYXYXY is thus a product of 10101 and the 2-digit number formed by XY (which is \(10X + Y\)). This means that XYXYXY is always divisible by 10101, regardless of the digits X and Y (as long as it forms a 6-digit number, meaning X is not 0).
Finding the Prime Factors of 10101
To find out the divisors of XYXYXY, we need to find the factors of 10101. Let's find the prime factorization of 10101:
- Check divisibility by 3: The sum of the digits of 10101 is \(1+0+1+0+1 = 3\). Since the sum of digits is divisible by 3, 10101 is divisible by 3.
- Divide 10101 by 3: \(10101 \div 3 = 3367\).
- Now let's check 3367 for other prime factors.
- Not divisible by 2 (it's odd).
- Not divisible by 3 (sum of digits \(3+3+6+7=19\), not divisible by 3).
- Not divisible by 5 (doesn't end in 0 or 5).
- Try 7: \(3367 \div 7 = 481\).
- Now let's check 481.
- Not divisible by 7 (\(481 = 7 \times 68 + 5\)).
- Not divisible by 11 (\(481 = 11 \times 43 + 8\)).
- Try 13: \(481 \div 13 = 37\).
- 37 is a prime number.
So, the prime factorization of 10101 is \(3 \times 7 \times 13 \times 37\).
Conclusion on Divisibility
Since \(\text{XYXYXY} = 10101 \times (10X + Y)\), and \(10101 = 3 \times 7 \times 13 \times 37\), we can write:
\(\text{XYXYXY} = (3 \times 7 \times 13 \times 37) \times (10X + Y)\)
This shows that any 6-digit number of the form XYXYXY is always divisible by 3, 7, 13, and 37. These are fixed divisors for any number of this specific form.
Comparing this result with the given options, we find that the number is divisible by 3, 7, 13 and 37.
Revision Table: Key Divisors of XYXYXY Numbers
| Number Form | Structure | Factored Form | Fixed Divisors |
|---|---|---|---|
| XYXYXY | \(101010X + 10101Y\) | \(10101 \times (10X+Y)\) | \(10101 = 3 \times 7 \times 13 \times 37\) |
Additional Information: Understanding Repeating Digit Patterns
Numbers with repeating digit patterns often have interesting divisibility properties based on the number 1, 101, 1001, 10101, etc.
- A number of the form ABAB (\(1010A + 10B\)) can be written as \(101(10A + B)\). It is always divisible by 101 (which is prime).
- A number of the form ABCABC (\(1000ABC + ABC\)) can be written as \(1001(100A + 10B + C)\). It is always divisible by \(1001 = 7 \times 11 \times 13\).
- The number XYXYXY form (\(10101 \times (10X + Y)\)) is divisible by \(10101 = 3 \times 7 \times 13 \times 37\).
These patterns arise because the structure of the number allows for factoring out numbers like 101, 1001, 10101, which have specific prime factorizations.
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select the correct answer using the code given below:
