A six digit number is formed by writing a three digit number twice in succession (for example, 404404 or 415415). Any number of this form is always exactly divisible by which of the following?

Introduction / Context:
This question deals with number patterns and divisibility properties. When a three digit number is repeated to form a six digit number, there is a special algebraic structure behind it. Recognising that structure allows us to discover a fixed divisor that always divides such a number, independent of the specific three digit chosen. This idea is useful in many number puzzles and aptitude tests.

Given Data / Assumptions:

• We take any three digit number, say ABC, where A, B, and C are digits and A is nonzero.
• We form a six digit number by writing ABC twice in succession, like ABCABC.
• Examples include 404404 and 415415.
• We must determine which given number always divides any such six digit number exactly.

Concept / Approach:
We translate the pattern ABCABC into an algebraic form using place values. If the three digit number is N, then the six digit number obtained by repeating N is N * 1000 + N, because shifting N left by three places multiplies it by 1000. That expression factorises as N * (1000 + 1) = N * 1001. This shows that every such number is a multiple of 1001. Then we simply check which option corresponds to this factor. Understanding place value and simple algebra is the key idea.

Step-by-Step Solution:
Step 1: Let the three digit number be N. For example, N could be 415.Step 2: When we write N twice, we get a six digit number. In general, this is N followed by N.Step 3: In decimal notation, writing N followed by three zeros corresponds to multiplying N by 1000, so the six digit number is N * 1000 + N.Step 4: Factor N * 1000 + N as N * (1000 + 1) = N * 1001.Step 5: This shows the six digit number is always a multiple of 1001, regardless of what the original N is.Step 6: Therefore each such number is exactly divisible by 1001.

Verification / Alternative check:
Take a concrete example, such as N = 415. The repeated number is 415415. Divide 415415 by 1001. Since 1001 = 7 * 11 * 13, you can check divisibility step by step. Performing the division or using a calculator confirms that 415415 ÷ 1001 = 415, giving zero remainder. Similarly, 404404 ÷ 1001 = 404. This confirms that 1001 divides all numbers of the given form and that the algebraic reasoning is correct.

Why Other Options Are Wrong:
Option 101: Although 1001 and 101 share some digits, 101 does not necessarily divide all numbers of the form ABCABC.Option 901: This number has no special algebraic relation to 1000 + 1 and will not always divide N * 1001.Option 789: This is unrelated to the structure of 1001 and is not a factor of 1001.Option 99: Often appears in repeating two digit patterns, but here the repetition is of three digits, leading to the factor 1001 instead.

Common Pitfalls:
Students sometimes misinterpret the pattern and think about the sum of digits or other properties, instead of thinking in terms of place value and algebra. Another common error is to assume that a smaller number like 101 or 99 is always the divisor without verifying algebraically. Remember to express the pattern in terms of N and standard base 10 powers, then factor systematically.

Final Answer:
Every such six digit number is always divisible by 1001.