Number System
🔒 Log in to trackDivisibility, remainders, unit digits, factors and recurring decimals — the base layer every other Quant topic uses. CGL asks 1–3 direct questions per Tier 1 shift (missing-digit divisibility, remainders and unit digit are near-certain) and a few more in Tier 2.
One page per subtopic: detailed notes, every question type, formulas, tricks and practice sets.
Every formula on one printable page, grouped by subtopic.
5 exam-level questions worked step by step.
97 questions — untimed practice or a timed test with analysis.
Track record in the exam
Questions per shift in recent SSC CGL papers.
Test difficulty mix (97 questions)
Question patterns exams keep repeating
Taken from previous-year papers. If a pattern is marked "very common", expect to see it in your exam.
Missing digit so a number divides evenly
very commonFind the missing digit in a number that must divide by 9, 11 or 88.
Use the divisibility rule for each part of the divisor.
Try only the few digits that fit.
Find x if 7x35 is divisible by 11.
Odd places: 7 + 3 = 10
Even places: x + 5
For 11, these must match (or differ by 11): x + 5 = 10
So x = 5
Remainder when the divisor is a factor of the old one
very common"A number divided by 195 leaves 47. What is the remainder when divided by 15?"
Check that the new divisor divides the old one.
Then divide the old remainder by the new divisor.
A number divided by 195 leaves 47. Remainder when divided by 15?
195 = 15 × 13, so 15 divides 195
So the remainder is the same as 47 ÷ 15
47 = 3 × 15 + 2
Remainder = 2
Remainder of a big power
very common"What is the remainder when 67^67 + 67 is divided by 68?"
Find a number close to the divisor, like 1 less, and treat it as −1.
An odd power of −1 is −1; an even power is +1.
Remainder of 67^67 + 67 divided by 68?
67 is 1 less than 68, so treat it as −1
(−1) to an odd power = −1
−1 + 67 = 66
Remainder = 66
Last digit of a big power
very common"Find the unit digit of 2137^754."
Keep only the last digit of the base.
Last digits repeat in a cycle of 4; divide the power by 4 and use the remainder.
Unit digit of 2137^754?
Last digit of the base = 7
Powers of 7 end in 7, 9, 3, 1 and repeat
754 ÷ 4 leaves 2, so take the 2nd: 9
Unit digit = 9
Count the factors of a number
very common"How many factors (or odd factors) does 3600 have?"
Break the number into primes.
Add 1 to each power and multiply. For odd factors, ignore the 2s.
How many odd factors does 3600 have?
3600 = 2⁴ × 3² × 5²
Ignore the 2s: the powers left are 2 and 2
(2 + 1) × (2 + 1) = 3 × 3
Odd factors = 9
Zeros at the end of a factorial
common"How many zeros are at the end of 125!?"
Count the 5s: divide by 5, then keep dividing the answer by 5.
Add all the answers.
How many zeros are at the end of 125!?
125 ÷ 5 = 25
25 ÷ 5 = 5
5 ÷ 5 = 1
25 + 5 + 1 = 31 zeros
Is this big sum or difference divisible by ...?
common"17^25 + 23^25 is divisible by which number?"
Odd power with a + sign: divisible by the sum of the two numbers.
Difference: divisible by the difference of the two numbers.
17^25 + 23^25 is divisible by which number?
The power 25 is odd and the sign is +
So it divides by 17 + 23
17 + 23 = 40
Answer: 40
Repeating decimals into fractions
common"Find 0.666... + 0.777... + 0.888..."
Put the repeating digit over 9: 0.777... = 7/9.
Change every decimal into a fraction, then add.
0.666... + 0.777... + 0.888... = ?
0.666... = 6/9
0.777... = 7/9
0.888... = 8/9
6/9 + 7/9 + 8/9 = 21/9 = 7/3 (2⅓)
Count multiples in a range
common"How many numbers from 1 to 1000 are divisible by neither 4 nor 6?"
Multiples of k up to N = N ÷ k (ignore the decimal part).
For 'both', use the LCM. Subtract the overlap once.
How many numbers from 1 to 1000 are divisible by neither 4 nor 6?
Multiples of 4: 1000 ÷ 4 = 250
Multiples of 6: 1000 ÷ 6 = 166
Both (multiples of 12): 1000 ÷ 12 = 83
Either: 250 + 166 − 83 = 333, so neither = 1000 − 333 = 667
Two-digit number and its reverse
occasional"A number is 4 times the sum of its digits. Adding 27 reverses the digits."
Call the digits a and b. The number is 10a + b.
Number and reverse differ by 9 × (difference of digits).
A two-digit number is 4 times its digit sum. Adding 27 reverses it. Find it.
10a + b = 4(a + b), so b = 2a
Reverse minus number = 9(b − a) = 27, so b − a = 3
So a = 3 and b = 6
Number = 36