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CSAT · 140 questions

Number system

Every UPSC CSAT question on this topic, 2016–2026, newest first. Tap an option to check yourself; the answer and explanation open below it.

Number system questions per year: 2016: 3, 2017: 7, 2018: 5, 2019: 13, 2020: 19, 2021: 10, 2022: 13, 2023: 21, 2024: 15, 2025: 22, 2026: 12 Asked in 11 of 11 years · most in 2025 (22)
Timed · 30 minutes a set Practise Number system as timed tests 7 sets of about 20 questions with UPSC's CSAT marking. Start practising

UPSC syllabus: “Basic numeracy (numbers and their relations, orders of magnitude, etc.) (Class X level), Data interpretation (charts, graphs, tables, data sufficiency etc. — Class X level);” Sub-topics in the detailed syllabus →

CSAT 2017 · Q35

X-Ray · Core #4 Easy

तेरह 2-अंकीय क्रमागत विषम संख्याएँ हैं। यदि ऐसी प्रथम पाँच संख्याओं का माध्य 39 है, तो सभी तेरह संख्याओं का माध्य क्या है?

Read this question in English

There are thirteen 2-digit consecutive odd numbers. If 39 is the mean of the first five such numbers, then what is the mean of all the thirteen numbers?

(a) 47 · (b) 49 · (c) 51 · (d) 45

उत्तर और व्याख्या · explanation in English

Answer: (a) 47

The mean of five consecutive odd numbers is the middle (third) one, so the third number is 39 and the list starts at 35. The mean of thirteen consecutive odd numbers is the 7th number: 35 + 6 × 2 = 47.

  1. Mean of 5 consecutive odd numbers = the 3rd number = 39, so the first number is 39 − 4 = 35.
  2. The 13 numbers are 35, 37, …, 59 (35 + 12 × 2 = 59), all 2-digit.
  3. Mean of an evenly spaced list = middle (7th) term = 35 + 6 × 2 = 47.
  4. Check: (35 + 59) ÷ 2 = 47.

Remember · For an evenly spaced list, the mean equals the middle term, or the average of the first and last terms.

Question and answer: UPSC's official GS Paper II (2017, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink ·

CSAT 2017 · Q49

Hard

कुछ 3-अंकीय संख्याओं की निम्नलिखित विशेषताएँ हैं :

  1. 1.सभी तीन अंक भिन्न-भिन्न हैं।
  2. 2.संख्या 7 से विभाजित होती है।
  3. 3.संख्या के अंकों को उलट देने से बनने वाली संख्या भी 7 से विभाजित होती है।

ऐसी कितनी 3-अंकीय संख्याएँ हो सकती हैं?

Read this question in English

Certain 3-digit numbers have the following characteristics:

1. All the three digits are different.

2. The number is divisible by 7.

3. The number on reversing the digits is also divisible by 7.

How many such 3-digit numbers are there?

(a) 2 · (b) 4 · (c) 6 · (d) 8

उत्तर और व्याख्या · explanation in English

Answer: (b) 4

If a number and its reverse are both multiples of 7, so is their difference, 99 × (first digit − last digit). Since 99 is not a multiple of 7, the first and last digits must differ by 7 — giving just 168, 861, 259 and 952.

  1. Let the number be 100a + 10b + c; its reverse is 100c + 10b + a.
  2. Their difference is 99(a − c). Both are multiples of 7 and 99 is not, so a − c must be a multiple of 7.
  3. The digits are all different, so a − c = ±7: (a, c) = (1, 8), (8, 1), (2, 9), (9, 2) or (7, 0). The last fails: 7b0 is divisible by 7 only if b = 0 or 7, which repeats a digit.
  4. Find the middle digit in each case: 1b8 → 168 = 7 × 24; 8b1 → 861 = 7 × 123; 2b9 → 259 = 7 × 37; 9b2 → 952 = 7 × 136.
  5. Each case has exactly one suitable middle digit, so there are 4 such numbers.
  6. Check: 861 and 952 are the reverses of 168 and 259, and all four have three different digits.

Remember · A 3-digit number and its reverse differ by 99 × (first − last digit); this turns 'both divisible by 7' into an end-digit condition.

Question and answer: UPSC's official GS Paper II (2017, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink ·

CSAT 2017 · Q51

X-Ray · Core #4 Easy

99 तथा 1000 के बीच ऐसी कितनी संख्याएँ हैं, जिनमें अंक 8 इकाई स्थान पर है?

Read this question in English

How many numbers are there between 99 and 1000 such that the digit 8 occupies the units place?

(a) 64 · (b) 80 · (c) 90 · (d) 104

उत्तर और व्याख्या · explanation in English

Answer: (c) 90

Numbers between 99 and 1000 are the 3-digit numbers 100 to 999. With 8 fixed in the units place, the hundreds digit has 9 choices and the tens digit 10 choices, giving 9 × 10 = 90.

  1. Numbers strictly between 99 and 1000: 100 to 999.
  2. Units digit fixed as 8.
  3. Hundreds digit: 1–9 (9 ways); tens digit: 0–9 (10 ways).
  4. Count = 9 × 10 = 90.
  5. Check: 108, 118, …, 998 rise in steps of 10, so the count is (998 − 108) ÷ 10 + 1 = 90.

Remember · Fix the given digit, then multiply the free choices for the other places — the leading digit cannot be 0.

Question and answer: UPSC's official GS Paper II (2017, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink ·

CSAT 2017 · Q53

X-Ray · Core #4 Medium

श्री X की आयु एक वर्ष पूर्व किसी संख्या का वर्ग थी तथा अगले वर्ष यह किसी संख्या का घन हो जाएगी। उसे अपनी आयु के पुनः किसी संख्या का घन होने के लिए न्यूनतम कितने वर्ष प्रतीक्षा करनी पड़ेगी?

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The age of Mr. X last year was the square of a number and it would be the cube of a number next year. What is the least number of years he must wait for his age to become the cube of a number again?

(a) 42 · (b) 38 · (c) 25 · (d) 16

उत्तर और व्याख्या · explanation in English

Answer: (b) 38

Next year's age is 2 more than last year's, so we need a square that is 2 less than a cube: 25 + 2 = 27. Mr. X was 25, is 26 now and turns 27 next year. The next cube is 64, which is 38 years away from his present age.

  1. If last year's age is n² and next year's is m³, then m³ − n² = 2.
  2. Try cubes: 8 − 2 = 6 (not a square), 27 − 2 = 25 = 5² ✓, 64 − 2 = 62, 125 − 2 = 123, 216 − 2 = 214 — only 27 works for a human age.
  3. So he was 25 last year, is 26 now and will be 27 next year.
  4. After 27, the next cube is 4³ = 64.
  5. Years to wait from his present age: 64 − 26 = 38.

Remember · 'Square last year, cube next year' means the two numbers differ by 2 — list squares and cubes side by side.

Question and answer: UPSC's official GS Paper II (2017, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink ·

CSAT 2017 · Q73

Medium

एक 2-अंकीय संख्या को उत्क्रमित किया गया। उन दो संख्याओं में से बड़ी संख्या को छोटी संख्या से विभाजित किया गया। वृहत्तम संभव शेषफल क्या है?

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A 2-digit number is reversed. The larger of the two numbers is divided by the smaller one. What is the largest possible remainder?

(a) 9 · (b) 27 · (c) 36 · (d) 45

उत्तर और व्याख्या · explanation in English

Answer: (d) 45

A number and its reverse differ by 9 × (difference of the digits). When the larger is less than twice the smaller, the quotient is 1 and the remainder is that difference; the biggest such case is 94 ÷ 49, which leaves 45.

  1. Let the larger number be 10a + b and the smaller 10b + a, with a > b.
  2. Their difference is 9(a − b). If the larger is less than twice the smaller, the quotient is 1 and the remainder is exactly 9(a − b).
  3. Quotient 1 needs 10a + b < 2(10b + a), i.e. 8a < 19b. With a = 9 this needs b ≥ 4, so the best pair is 94 and 49: remainder 9 × 5 = 45.
  4. If the quotient is 2 or more, then 8a ≥ 19b forces b ≤ 3, so the smaller number is at most 39 and the remainder is less than 39.
  5. Largest possible remainder = 45 (94 = 1 × 49 + 45).

Remember · A number and its reverse differ by 9 × (difference of digits); when the quotient is 1, that difference is the remainder.

Question and answer: UPSC's official GS Paper II (2017, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink ·

CSAT 2017 · Q79

Easy

कुछ 2-अंकीय संख्याएँ हैं। इन संख्याओं और इनके अंकों को उलट देने पर बनने वाली संख्याओं का अंतर सदैव 27 रहता है। ऐसी अधिकतम कितनी 2-अंकीय संख्याएँ हैं?

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There are certain 2-digit numbers. The difference between the number and the one obtained on reversing it is always 27. How many such maximum 2-digit numbers are there?

(a) 3 · (b) 4 · (c) 5 · (d) None of the above

उत्तर और व्याख्या · explanation in English

Answer: (d) उपर्युक्त में से कोई नहीं

A 2-digit number and its reverse differ by 9 × (difference of the digits), so the digits must differ by 3. That gives 41, 52, 63, 74, 85, 96 and 14, 25, 36, 47, 58, 69 (and 30, if 03 is allowed) — at least 12 numbers, far more than 3, 4 or 5.

  1. (10a + b) − (10b + a) = 9(a − b), which is 27 in size when the digits differ by 3.
  2. Tens digit 3 more than units: 41, 52, 63, 74, 85, 96 (and 30, if its reverse 03 = 3 is accepted).
  3. Tens digit 3 less than units: 14, 25, 36, 47, 58, 69.
  4. That is 12 numbers (13 counting 30); even one direction alone gives 6. None of 3, 4 or 5 fits.

Remember · Number − reverse = 9 × (digit difference). Divide the given difference by 9 to get the digit gap, then list the pairs.

Question and answer: UPSC's official GS Paper II (2017, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink ·

CSAT 2017 · Q80

X-Ray · Core #4 Easy

यदि 150 पृष्ठों की एक पुस्तक में 1 से 150 तक संख्याएँ अंकित करनी हैं, तो पुस्तक में मुद्रित अंकों की कुल संख्या क्या है?

Read this question in English

What is the total number of digits printed, if a book containing 150 pages is to be numbered from 1 to 150?

(a) 262 · (b) 342 · (c) 360 · (d) 450

उत्तर और व्याख्या · explanation in English

Answer: (b) 342

Pages 1–9 use 9 digits, pages 10–99 use 90 × 2 = 180 digits, and pages 100–150 use 51 × 3 = 153 digits. The total is 342.

  1. Pages 1–9: 9 pages × 1 digit = 9.
  2. Pages 10–99: 90 pages × 2 digits = 180.
  3. Pages 100–150: 51 pages × 3 digits = 153.
  4. Total = 9 + 180 + 153 = 342.

Remember · Count digits band by band: pages in a band = last − first + 1, times the digits on each page.

Question and answer: UPSC's official GS Paper II (2017, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). Permalink ·

The Syllabus X-Ray of this topic

1 of its 5 sub-topics is Core

1 Core · 2 High · 2 Steady · 0 Rare

  • Core #4Number properties & sequences · 70 questions
  • HighDivisibility, remainders, HCF & LCM · 34 questions
  • HighSequences, cycles & extreme values · 30 questions

The X-Ray ranks all 928 sub-topics of the UPSC syllabus by what UPSC has asked since 2016. The 6 CSAT Core sub-topics are behind 50% of CSAT questions.

See the whole X-Ray →

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