Numbers— Concepts, Formulas & Shortcuts
- Divisibility: by 3 or 9 → digit sum; by 4 → last two digits; by 8 → last three digits; by 11 → alternating digit sum.
- Unit digits cycle with period 4: for 3ⁿ the cycle is 3, 9, 7, 1 — take n mod 4.
- Sum of the first n natural numbers = n(n + 1)/2; sum of squares = n(n + 1)(2n + 1)/6.
- A number divisible by two co-prime numbers is divisible by their product (5 and 8 ⇒ 40).
- Remainder shortcut: reduce the base modulo the divisor first, e.g. 17 ≡ −1 (mod 18).
- Primes below 100 that are commonly tested: 61, 67, 71, 73, 79, 83, 89, 97.
Numbers Practice Questions with Answers
Attempt each question first, then open the explanation. All 24 questions below are free to read and require no signup.
Q1.The unit digit of 3⁶⁵ is:
Moderate- A1
- B3
- C7
- D9
Numbers question 1 of 24+Show Answer & Explanation
Answer: B. 3
Explanation: The cycle of 3 is 3, 9, 7, 1 with period 4. 65 mod 4 = 1 → unit digit = 3.
Q2.Which of the following is a prime number?
Easy- A87
- B91
- C93
- D97
Numbers question 2 of 24+Show Answer & Explanation
Answer: D. 97
Explanation: 87 = 3 × 29, 91 = 7 × 13, 93 = 3 × 31; 97 has no factor up to 9, so it is prime.
Q3.Which of these numbers is divisible by 9?
Easy- A12,345
- B45,927
- C52,341
- D61,208
Numbers question 3 of 24+Show Answer & Explanation
Answer: B. 45,927
Explanation: Digit sum of 45,927 = 4 + 5 + 9 + 2 + 7 = 27, which is divisible by 9.
Q4.The largest 4-digit number exactly divisible by 88 is:
Moderate- A9,944
- B9,768
- C9,988
- D9,900
Numbers question 4 of 24+Show Answer & Explanation
Answer: A. 9,944
Explanation: 9999 ÷ 88 = 113 remainder 55 → 113 × 88 = 9,944.
Q5.The sum of the first 50 natural numbers is:
Easy- A1,175
- B1,250
- C1,275
- D1,375
Numbers question 5 of 24+Show Answer & Explanation
Answer: C. 1,275
Explanation: n(n + 1)/2 = 50 × 51/2 = 1,275.
Q6.How many numbers between 1 and 100 are divisible by both 3 and 5?
Easy- A5
- B6
- C7
- D8
Numbers question 6 of 24+Show Answer & Explanation
Answer: B. 6
Explanation: Multiples of 15: 15, 30, 45, 60, 75, 90 → 6 numbers.
Q7.A number divisible by both 5 and 8 must also be divisible by:
Easy- A13
- B20
- C40
- D45
Numbers question 7 of 24+Show Answer & Explanation
Answer: C. 40
Explanation: 5 and 8 are co-prime, so the number is divisible by 5 × 8 = 40.
Q8.The remainder when 17²⁰⁰ is divided by 18 is:
Difficult- A0
- B1
- C16
- D17
Numbers question 8 of 24+Show Answer & Explanation
Answer: B. 1
Explanation: 17 ≡ −1 (mod 18), so 17²⁰⁰ ≡ (−1)²⁰⁰ = 1.
Q9.The unit digit of 7¹⁰⁰ is:
Difficult- A1
- B3
- C7
- D9
Numbers question 9 of 24+Show Answer & Explanation
Answer: A. 1
Explanation: The cycle of 7 is 7, 9, 3, 1 with period 4. 100 mod 4 = 0, so the unit digit is the 4th in the cycle: 1.
Q10.Which of the following is divisible by 11?
Difficult- A12,321
- B45,678
- C90,728
- D83,919
Numbers question 10 of 24+Show Answer & Explanation
Answer: C. 90,728
Explanation: For 90,728 the alternating sums are (9 + 7 + 8) = 24 and (0 + 2) = 2, giving a difference of 22, a multiple of 11.
Q11.The sum of the first 20 odd natural numbers is:
Moderate- A200
- B380
- C400
- D420
Numbers question 11 of 24+Show Answer & Explanation
Answer: C. 400
Explanation: The sum of the first n odd numbers is n² = 20² = 400.
Q12.How many prime numbers are there between 1 and 30?
Moderate- A8
- B9
- C10
- D11
Numbers question 12 of 24+Show Answer & Explanation
Answer: C. 10
Explanation: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 → 10 primes.
Q13.The smallest 4-digit number divisible by 12 is:
Moderate- A1,000
- B1,002
- C1,008
- D1,012
Numbers question 13 of 24+Show Answer & Explanation
Answer: C. 1,008
Explanation: 1000 ÷ 12 leaves a remainder of 4, so the next multiple is 1000 + 8 = 1,008.
Q14.A number divisible by 4 must have its last two digits divisible by:
Easy- A2
- B4
- C8
- D16
Numbers question 14 of 24+Show Answer & Explanation
Answer: B. 4
Explanation: The divisibility rule for 4 depends only on the number formed by the last two digits.
Q15.The sum of the squares of the first 10 natural numbers is:
Moderate- A285
- B385
- C485
- D585
Numbers question 15 of 24+Show Answer & Explanation
Answer: B. 385
Explanation: n(n+1)(2n+1)/6 = 10 × 11 × 21/6 = 385.
Q16.The remainder when 2¹⁰ is divided by 7 is:
Difficult- A1
- B2
- C4
- D6
Numbers question 16 of 24+Show Answer & Explanation
Answer: B. 2
Explanation: 2³ = 8 ≡ 1 (mod 7), so 2¹⁰ = (2³)³ × 2 ≡ 1 × 2 = 2. Check: 1024 = 146 × 7 + 2.
Q17.The unit digit of 2⁵⁰ is:
Difficult- A2
- B4
- C6
- D8
Numbers question 17 of 24+Show Answer & Explanation
Answer: B. 4
Explanation: The cycle of 2 is 2, 4, 8, 6 with period 4. 50 mod 4 = 2 → the second in the cycle, 4.
Q18.Which of the following is divisible by 3?
Easy- A1,234
- B2,345
- C3,456
- D4,567
Numbers question 18 of 24+Show Answer & Explanation
Answer: C. 3,456
Explanation: Digit sum of 3,456 = 18, which is divisible by 3.
Q19.The sum of the first 10 natural numbers is:
Easy- A45
- B50
- C55
- D60
Numbers question 19 of 24+Show Answer & Explanation
Answer: C. 55
Explanation: n(n + 1)/2 = 10 × 11/2 = 55.
Q20.How many two-digit numbers are divisible by 7?
Moderate- A12
- B13
- C14
- D15
Numbers question 20 of 24+Show Answer & Explanation
Answer: B. 13
Explanation: Multiples of 7 from 14 to 98: (98 − 14)/7 + 1 = 13.
Q21.The smallest prime number is:
Easy- A0
- B1
- C2
- D3
Numbers question 21 of 24+Show Answer & Explanation
Answer: C. 2
Explanation: 2 is the smallest prime and the only even one; 1 is not prime.
Q22.A number divisible by 6 must be divisible by:
Easy- A4 and 3
- B2 and 3
- C2 and 4
- D3 and 5
Numbers question 22 of 24+Show Answer & Explanation
Answer: B. 2 and 3
Explanation: 6 = 2 × 3, so the number must be divisible by both 2 and 3.
Q23.The largest 3-digit number divisible by 9 is:
Moderate- A990
- B993
- C996
- D999
Numbers question 23 of 24+Show Answer & Explanation
Answer: D. 999
Explanation: 999 has digit sum 27, which is divisible by 9.
Q24.The number of factors of 36 is:
Difficult- A6
- B8
- C9
- D12
Numbers question 24 of 24+Show Answer & Explanation
Answer: C. 9
Explanation: 36 = 2² × 3², so the factor count is (2 + 1)(2 + 1) = 9.
Numbers — Frequently Asked Questions
How do I find the unit digit of a large power?+
Unit digits repeat with a cycle of at most 4. Divide the exponent by 4 and use the remainder: for 3⁶⁵, 65 mod 4 = 1, so the unit digit matches 3¹ = 3.
What is the quickest divisibility test for 11?+
Add alternate digits and subtract the two sums. If the result is 0 or a multiple of 11, the number is divisible by 11.
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