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Aptitude Topics

Divisibility Applications

Divisibility problems use LCM rules to identify numbers that leave specific remainders when divided by a given set of divisors.

Fundamental Principles

Uniform Remainder Rule

Finding the smallest target number that leaves a uniform remainder 'r' when divided by divisors a, b, and c. The solution follows the format: $\text{Target} = \text{LCM}(a, b, c) + r$.

Constant Difference Remainder Rule

When dividing by a, b, and c leaves different remainders $r_1, r_2, r_3$ but shares a constant difference $k$ such that $(a - r_1) = (b - r_2) = (c - r_3) = k$. The solution format is: $\text{Target} = \text{LCM}(a, b, c) - k$.

Essential Formulation Tips

  • Always check if the problem asks for a uniform positive remainder addition or a balanced factor step subtraction.
  • When looking for numbers near a specific ceiling limit, scale up your baseline LCM by integer multiples ($2k, 3k$) before processing any remainder math.

Shortcut Execution Techniques

  • To find the largest 4-digit number divisible by a set of numbers, divide 9999 by their LCM, find the remainder, and subtract that remainder from 9999.

Contextual Inquiries (FAQs)

Q: Why do we subtract the constant difference 'k' in varying remainder problems?

A: Because subtracting 'k' leaves the target number exactly short of completing the next full division step for each divisor, which generates the required matching remainders.