Divisibility Rules Practice Questions with Detailed Solutions – Level 3

Divisibility Rules Level 3 Practice Quiz: Missing Digits and Remainders

Aptitude360's Divisibility Rules Level 3 set is for quantitative aptitude learners who already know the basic tests and want hard Number System practice. It covers missing-digit problems for composite divisors, remainders by alternating and grouped-digit methods, and numbers made by repeating a digit or block. Every question comes with a worked solution.

Topics covered in this hard divisibility set

• finding missing digits when a number must be divisible by a composite divisor
• counting digit pairs that satisfy a divisibility condition
• remainders using the alternating digit sum, the pair-sum method and the triplet method
• numbers formed by repeating a single digit or a block of digits, including repunits
• palindromes and numbers made by repeating a three-digit block
• least number to add or subtract for divisibility by several divisors at once

Key divisibility rules and formulas

• Alternating-sum rule for eleven: \( N \equiv a_{0}-a_{1}+a_{2}-a_{3}+\cdots \pmod{11} \)
• Pair-sum rule for ninety-nine: \( 100 \equiv 1 \pmod{99} \)
• Triplet rule for thirty-seven: \( 1000 \equiv 1 \pmod{37} \)
• Factors of one thousand and one: \( 1001 = 7\times 11\times 13 \)
• Repeated three-digit block: \( \overline{abcabc} = \overline{abc}\times 1001 \)
• Splitting a composite divisor: \( 72 = 8\times 9,\ \gcd(8,9)=1 \)

Revision tips for advanced divisibility problems

• Split a composite divisor into co-prime factors and apply each rule separately.
• When the factors are a power of two and nine, settle the last-digits test first and then the digit-sum test; for other factors use their own rules, such as the alternating sum for eleven.
• For repeated blocks, find how many blocks form a multiple of the divisor and reduce the rest.
• Group digits in threes from the right with alternating signs to test for seven, eleven and thirteen in one step.

Who should practise this Level 3 set

It suits CAT, IPMAT, SSC and banking aspirants in the final stage of Number System preparation who are already comfortable with the basic divisibility tests.