1. Number system and divisibility
Free study material · concepts, shortcuts & solved questions
Start with place value, factors, multiples, prime factorisation, HCF and LCM. For two positive integers, product = HCF × LCM. Divisibility tests save time: 2 and 5 use the last digit; 3 and 9 use the digit sum; 4 uses the last two digits; 8 uses the last three digits; 11 uses the alternating sum.
Method before formula For HCF, take the minimum powers of common primes; for LCM, take the maximum powers. In word problems, decide whether the condition is “greatest measure” (HCF) or “first common occurrence” (LCM). |
Formula and decision table
Rule | Example |
HCF/LCM | 36=2²×3², 48=2⁴×3; HCF=12, LCM=144 |
Remainder | If N leaves r on division by d, N=qd+r |
Divisibility by 11 | Difference between alternating digit sums is 0 or a multiple of 11 |
Worked understanding
A reliable solution has five visible moves: identify the data, choose the base or unit, write the rule, substitute carefully, and check the size of the answer. The examples in this chapter are deliberately small so that the method remains visible.
Calculation discipline Before pressing ahead, state the denominator or unit in words. For example: profit on CP, discount on MP, average over number of items, speed in metres per second, and probability over the fixed sample space. |
Solved examples from this chapter
Question focus | Correct move | Answer |
HCF of 84 and 126 is: | 84=2²×3×7 and 126=2×3²×7; common minimum product is 42. | C. 42 |
LCM is used when events must occur: | LCM gives the first common multiple. | A. At the same first future time |
Which number is divisible by 9? | 7+2+9=18, divisible by 9. | B. 729 |
What an examiner is testing
The options around Number system and divisibility usually represent a wrong base, a missed conversion, a sign error or a shortcut used outside its condition. Say the base and unit before calculating, estimate the result, and use substitution or a reverse operation as the final check.
Step-by-step answer construction
Example 1: HCF of 84 and 126 is: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 42, because 84=2²×3×7 and 126=2×3²×7; common minimum product is 42.
Example 2: LCM is used when events must occur: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is At the same first future time, because LCM gives the first common multiple.
Example 3: Which number is divisible by 9? First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 729, because 7+2+9=18, divisible by 9.
Chapter practice
1. HCF of 84 and 126 is:
(A) 14 (B) 21 (C) 42 (D) 63
Answer: C. 42 | Explanation: 84=2²×3×7 and 126=2×3²×7; common minimum product is 42.
2. LCM is used when events must occur:
(A) At the same first future time (B) In the greatest possible size (C) Only once (D) With no common multiple
Answer: A. At the same first future time | Explanation: LCM gives the first common multiple.
3. Which number is divisible by 9?
(A) 734 (B) 729 (C) 721 (D) 725
Answer: B. 729 | Explanation: 7+2+9=18, divisible by 9.
SSC CGL speed and trap clinic
For Number system and divisibility, speed comes after classification. Before calculating, say aloud what the number means, what the denominator is, and what unit the answer must carry. Then use the nearest-option check to catch a sign, base or conversion error.
Checkpoint | What to verify | Typical SSC mistake |
Base | Which quantity is the base in Number system and divisibility? | Using selling price instead of cost price, or part instead of total |
Unit | Are time, distance, area and rate in compatible units? | Mixing hours with minutes or km/h with m/s |
Direction | Should the answer increase, decrease or stay bounded? | Accepting an impossible negative length or probability above 1 |
Estimate | What range should the answer lie in before exact work? | Trusting a long calculation that is far from the options |
Reverse check | Can the answer be substituted back into the condition? | Stopping at an algebraic value without verification |
Mini decision drill
1. Write the first operation you would perform in a Number system and divisibility question and why.
2. State the most dangerous denominator or unit in this chapter.
3. Give one condition under which a shortcut would be invalid.
4. Create a small numerical example and verify it by a second method.
Revision evidence Do not tick this chapter because you recognised the formula. Tick it only after you solve one direct question, one altered-condition question and one mixed-paper question correctly. |