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Quantitative Aptitudeमात्रात्मक योग्यता · संख्या पद्धति

Number Systemसंख्या पद्धति questions and answersप्रश्न और उत्तर

Number system is the most frequent SSC quant topic. Its questions look scary (huge powers, giant numbers) but almost all fall to a few cycles and rules.

  • Unlimitedअसीमित fresh questionsनए प्रश्न
  • Answersउत्तर always one tap awayहमेशा एक टैप दूर
  • 3 subtopicsउप-टॉपिक
  • SSC CGL ~3
  • TCS NQT ~2

Practice questions are in English for now.अभ्यास के प्रश्न अभी अंग्रेज़ी में हैं।

How number system worksसंख्या पद्धति को समझिए

Unit digits repeat in short cycles. Divide the power by 4, use the remainder (0 means 4), and read the unit digit from the cycle.

Divisibility rules let you test big numbers without dividing: digit sums for 3 and 9, alternating sums for 11, last digits for 2, 4, 5 and 8.

For remainders, replace each number by its remainder first. Products, sums and powers can all be reduced this way.

Tricks that save timeसमय बचाने वाली ट्रिक्स

All tricks →सभी ट्रिक्स →

Each trick says when to use it, why it works and when it fails. Formula tricks are checked by code over their whole range.हर ट्रिक बताती है कि उसे कब इस्तेमाल करें, वह क्यों काम करती है और कब नहीं। सूत्र वाली ट्रिक्स को कोड से उनकी पूरी रेंज पर जाँचा गया है।

Option killerविकल्प-छँटनीRarely taughtकम सिखाई जाने वाली✓ Checked on 6,560 cases✓ 6,560 मामलों पर जाँची गई

Option killer: the digit-sum check

Use it when:कब इस्तेमाल करें: Exact multiplication, addition or subtraction questions where all the options are close.

  1. Reduce each number to its digit sum, repeating until one digit (4,327 → 16 → 7).
  2. Do the same operation on these single digits and reduce again.
  3. The correct option must have the same digit sum. Usually only one does.
Exampleउदाहरण

347 × 26 = ? (a) 9,012 (b) 9,022 (c) 9,032 (d) 9,122

→ 347 → 5, 26 → 8, 5 × 8 = 40 → 4. Only 9,022 (9 + 0 + 2 + 2 = 13 → 4) matches.

Why it works, and when it failsयह क्यों काम करती है, और कब नहीं

Why:क्यों: A number and its digit sum leave the same remainder when divided by 9, and remainders survive addition, subtraction and multiplication (“casting out nines”).

Limits:सीमाएँ: Cannot tell apart options that differ by a multiple of 9, and does not work for division or decimals. Pair it with the last-digit check.

Option killerविकल्प-छँटनीRarely taughtकम सिखाई जाने वाली✓ Checked on 18,172 cases✓ 18,172 मामलों पर जाँची गई

Option killer: the last-digit check

Use it when:कब इस्तेमाल करें: Products, squares or sums where the options end in different digits.

  1. Multiply (or add) only the last digits of the numbers.
  2. Keep the last digit of that result; strike out every option that ends differently.
Exampleउदाहरण

67 × 43 = ? (a) 2,876 (b) 2,881 (c) 2,883 (d) 2,889

→ 7 × 3 = 21, so the product ends in 1. Only 2,881 ends in 1: that is the answer, without multiplying.

Why it works, and when it failsयह क्यों काम करती है, और कब नहीं

Why:क्यों: The last digit of a sum or product depends only on the last digits of the inputs (arithmetic mod 10).

Limits:सीमाएँ: Useless when all options share a last digit. Then use the digit-sum check, which catches different errors.

Mental mathsमानसिक गणितRarely taughtकम सिखाई जाने वाली✓ Checked on 600 cases✓ 600 मामलों पर जाँची गई

Last two digits of any power of a number ending in 1

Use it when:कब इस्तेमाल करें: “Find the last two digits of 41^27” or “the tens digit of 71^n”.

  1. The last digit is 1.
  2. Tens digit = last digit of (tens digit of the base × the power).
Exampleउदाहरण

Last two digits of 41^27

→ 4 × 27 = 108 → 8, so 81

Why it works, and when it failsयह क्यों काम करती है, और कब नहीं

Why:क्यों: (10t + 1)^n = 1 + n·10t + (terms with 100 or more). Mod 100 only 1 + 10·(n·t) survives.

Limits:सीमाएँ: Only for bases ending in 1. For other bases, first raise to a power that ends in 1 (for example 3⁴ = 81, 7⁴ = 2401, 9² = 81).

Mental mathsमानसिक गणितRarely taughtकम सिखाई जाने वाली✓ Checked on 138 cases✓ 138 मामलों पर जाँची गई

Test 7, 11 and 13 at once with 1001

Use it when:कब इस्तेमाल करें: Is a large number divisible by 7, 11 or 13? Or what is its remainder?

  1. Split the number into groups of three digits from the right.
  2. Alternately add and subtract the groups (rightmost +).
  3. The result has the same remainder as the original for 7, 11 and 13.
Exampleउदाहरण

Is 1,234,569 divisible by 7?

→ 569 − 234 + 1 = 336 = 7 × 48 → yes

Why it works, and when it failsयह क्यों काम करती है, और कब नहीं

Why:क्यों: 1000 ≡ −1 (mod 1001), and 1001 = 7 × 11 × 13, so each group of three digits flips sign.

Limits:सीमाएँ: Gives the remainder up to sign: if the alternating sum is negative, add 1001 first.

Formulaसूत्र✓ Checked on 1,005 cases✓ 1,005 मामलों पर जाँची गई

Same remainder → HCF of the differences

Use it when:कब इस्तेमाल करें: “Find the greatest number that divides 43, 91 and 183 leaving the same remainder.”

  1. Subtract the numbers pairwise: 91 − 43, 183 − 91.
  2. The answer is the HCF of the differences.
Exampleउदाहरण

43, 91, 183

→ HCF(48, 92) = 4

Why it works, and when it failsयह क्यों काम करती है, और कब नहीं

Why:क्यों: If a and b leave the same remainder when divided by d, then d divides b − a; the largest such d is the HCF of all the differences.

Limits:सीमाएँ: Only for “the same remainder”. For different given remainders, subtract each remainder from its number first.

Formulaसूत्र✓ Checked on 600 cases✓ 600 मामलों पर जाँची गई

Number of digits = ⌊log N⌋ + 1

Use it when:कब इस्तेमाल करें: “How many digits does 2⁵⁰ (or 3⁴⁰, or 7²⁰) have?” with log 2, log 3 or log 7 given.

  1. Find log N = power × log of the base (log 2 ≈ 0.30103, log 3 ≈ 0.47712, log 7 ≈ 0.84510).
  2. Drop the decimal part and add 1.
Exampleउदाहरण

Digits in 2⁵⁰?

→ log 2⁵⁰ = 50 × 0.30103 = 15.05 → 15 + 1 = 16 digits.

Why it works, and when it failsयह क्यों काम करती है, और कब नहीं

Why:क्यों: A number with d digits lies between 10ᵈ⁻¹ and 10ᵈ, so its log lies in [d − 1, d). The whole part of log N is therefore d − 1.

Limits:सीमाएँ: Needs enough decimal places in the given log: if the product lands very close to a whole number, four-figure logs may not be accurate enough.

Formulaसूत्र✓ Checked on 90 cases✓ 90 मामलों पर जाँची गई

Reversing a two-digit number: difference is 9 × gap, sum is 11 × digit sum

Use it when:कब इस्तेमाल करें: “When the digits of a two-digit number are reversed, the number increases (or decreases) by …”, or the number plus its reverse is given.

  1. Divide the change by 9: that is the difference between the two digits.
  2. Divide a given “number + reverse” by 11: that is the sum of the digits.
  3. Sum and difference of the digits fix them; an increase on reversing means the smaller digit is in the tens place.
Exampleउदाहरण

Digit sum 11; reversing the number decreases it by 45.

→ Gap = 45/9 = 5, sum 11 → digits 8 and 3. It decreases, so the larger digit is first: 83.

Why it works, and when it failsयह क्यों काम करती है, और कब नहीं

Why:क्यों: (10a + b) − (10b + a) = 9(a − b) and (10a + b) + (10b + a) = 11(a + b).

Limits:सीमाएँ: Two-digit numbers only. For three digits, reversing changes the number by 99 × (first digit − last digit).

Solved examplesहल किए गए उदाहरण

Answers and full solutions are shown. Hide them to test yourself first.उत्तर और पूरे हल दिखाए गए हैं। पहले खुद को परखने के लिए इन्हें छिपाइए।

  1. Example 1उदाहरण 1 · Unit digit of large powers · Easyआसान

    What is the unit digit of 14^287?

    1. 0
    2. 4
    3. 7
    4. 3
    Answer:उत्तर: 4
    1. 14 ends in 4; 287 ÷ 4 leaves remainder 3, so use power 3: 4^3 ends in 4.

  2. Example 2उदाहरण 2 · Divisibility rules: find the missing digit · Easyआसान

    What is the smallest digit that can replace * so that 9*71353 is divisible by 9?

    1. 6
    2. 7
    3. 8
    4. 9
    Answer:उत्तर: 8
    1. Digit sum must be divisible by 9.
    2. Digits that work: 8.
    3. Smallest = 8: 98,71,353 ÷ 9 = 10,96,817.

  3. Example 3उदाहरण 3 · Remainders · Easyआसान

    A number when divided by 99 leaves remainder 42. What is the remainder when the same number is divided by 11?

    1. 7
    2. 9
    3. 10
    4. 11
    Answer:उत्तर: 9
    1. N = 99k + 42, and 99k is divisible by 11.
    2. 42 ÷ 11 leaves 9.

  4. Example 4उदाहरण 4 · Unit digit of large powers · Mediumमध्यम

    What is the unit digit of 9^278 × 99^257?

    1. 8
    2. 9
    3. 5
    4. 2
    Answer:उत्तर: 9
    1. 9 ends in 9; 278 ÷ 4 leaves remainder 2, so use power 2: 9^2 ends in 1.
    2. 99 ends in 9; 257 ÷ 4 leaves remainder 1, so use power 1: 9^1 ends in 9.
    3. 1 × 9 = 9 → unit digit 9.

  5. Example 5उदाहरण 5 · Divisibility rules: find the missing digit · Mediumमध्यम

    What is the smallest digit that can replace * so that 869*0 is divisible by 6?

    1. 1
    2. 2
    3. 3
    4. 4
    Answer:उत्तर: 1
    1. Divisible by 6 = divisible by both 2 and 3.
    2. Digits that work: 1, 4, 7.
    3. Smallest = 1: 86,910 ÷ 6 = 14,485.

  6. Example 6उदाहरण 6 · Remainders · Mediumमध्यम

    What is the remainder when 80 × 55 × 73 is divided by 7?

    1. 4
    2. 5
    3. 6
    4. 8
    Answer:उत्तर: 5
    1. Remainders on dividing by 7: 3, 6, 3.
    2. 3 × 6 × 3 = 54.
    3. 54 ÷ 7 leaves 5.

  7. Example 7उदाहरण 7 · Unit digit of large powers · Hardकठिन

    What is the unit digit of 43^266 + 9^143?

    1. 1
    2. 7
    3. 4
    4. 8
    Answer:उत्तर: 8
    1. 43 ends in 3; 266 ÷ 4 leaves remainder 2, so use power 2: 3^2 ends in 9.
    2. 9 ends in 9; 143 ÷ 4 leaves remainder 3, so use power 3: 9^3 ends in 9.
    3. 9 + 9 → unit digit 8.

  8. Example 8उदाहरण 8 · Divisibility rules: find the missing digit · Hardकठिन

    What is the smallest digit that can replace * so that 72*464 is divisible by 36?

    1. 2
    2. 3
    3. 4
    4. 5
    Answer:उत्तर: 4
    1. 36 = 4 × 9: the number must pass both tests.
    2. Digits that work: 4.
    3. Smallest = 4: 7,24,464 ÷ 36 = 20,124.

Common mistakesआम गलतियाँ

  • Using remainder 0 instead of the 4th term of the cycle.
  • Applying the divisibility rule for 11 from the wrong end (either end works if you stay consistent).

Frequently asked questionsअक्सर पूछे जाने वाले प्रश्न

How do I find the unit digit of 7^95?

95 ÷ 4 leaves 3, and the cycle of 7 is 7, 9, 3, 1, so the unit digit is 3.

What is the divisibility rule for 11?

The difference between the sums of alternate digits must be 0 or a multiple of 11.

Is number system important for SSC CGL?

Yes. It is among the most asked quant topics in SSC papers.

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