Divisibility of 6-digit number XYZXYZ
Question
For any choices of values of X, Y and Z, the 6-digit number of the form XYZXYZ is divisible by:
Options
7 and 11 only
11 and 13 only
7 and 13 only
7, 11 and 13
Explanation
Expand the 6-digit repeating pattern using its place-value representation: XYZXYZ = 100000X + 10000Y + 1000Z + 100X + 10Y + Z Group matching variables: = 100100X + 10010Y + 1001Z = 1001(100X + 10Y + Z) = 1001 × (XYZ).
The structural multiplier of any repeating 3-digit pattern block is exactly 1001. Find the prime factorization of 1001: 1001 = 7 × 11 × 13. Since 1001 is a product of 7, 11, and 13, any number formatted as XYZXYZ is universally divisible by all three numbers.
Answer: (d).
Question details
Year
2023
Paper
CSAT
Question
Q17
Section
Numerical Ability
Sub-topic
Divisibility Rules
Type
Number theory
Difficulty
Easy
Source hint
Number theory
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