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Classification, integers, even-odd, and prime-composite logic.
Divisibility rules, digit logic, and why common tests work.
Factors, multiples, prime factorisation, and factor counting.
Remainders, cyclicity, unit digits, and quick modular thinking.
A mixed mock covering all Number System subtopics. Ideal after reading the notes and trying the solved examples.
1. Which of the following is a whole number but not a natural number?
Explanation: Natural numbers begin from 1, while whole numbers begin from 0. So 0 belongs to whole numbers, but it is not a natural number.
2. Which of the following is an irrational number?
Explanation: A rational number can be written in the form $p/q$. $3/7$, 0.25 and -8 are all rational. $\sqrt{5}$ is non-terminating and non-repeating, so it is irrational.
3. If one number is even and another is odd, then their sum is:
Explanation: Even + Odd = Odd. Example: 6 + 5 = 11, which is odd.
4. Which is the smallest prime number?
Explanation: A prime number has exactly two distinct positive factors. 2 has factors 1 and 2 only, so it is prime. It is also the only even prime number.
5. Which statement is true about the number 1?
Explanation: A prime number must have exactly two distinct factors. The number 1 has only one positive factor, namely 1, so it is neither prime nor composite.
6. Which of the following is the greatest?
Explanation: Among negative numbers, the number closer to 0 is greater. So among -9, -3, -11 and -6, the greatest is -3.
7. Which of the following is a rational number?
Explanation: 0.375 is a terminating decimal, so it can be written as a fraction: $0.375 = 375/1000 = 3/8$. Hence it is rational.
8. If n is an odd integer, then n^2 is:
Explanation: Let $n = 2k+1$. Then $(2k+1)^2 = 4k^2 + 4k + 1 = 2(2k^2+2k)+1$, which is of the form $2m+1$. So it is always odd.
9. How many prime numbers lie between 10 and 20?
Explanation: Prime numbers between 10 and 20 are 11, 13, 17 and 19. Total = 4.
10. Which set correctly represents integers?
Explanation: Integers include negative whole numbers, zero, and positive whole numbers. So the correct set is $\{..., -2, -1, 0, 1, 2, ...\}$.
11. The number 738 is divisible by:
Explanation: Sum of digits = 7 + 3 + 8 = 18. Since 18 is divisible by 3 and also by 9, 738 is divisible by both.
12. 4516 is divisible by:
Explanation: For divisibility by 4, last two digits are 16, which is divisible by 4. For divisibility by 8, last three digits are 516, and 516 is not divisible by 8.
13. The smallest digit that can replace x in 43x2 so that the number is divisible by 3 is:
Explanation: Sum of digits = 4 + 3 + x + 2 = 9 + x. For divisibility by 3, $9 + x$ must be divisible by 3. So x can be 0, 3, 6, or 9. The smallest is 0.
14. The greatest digit that can replace y in 57y4 so that the number is divisible by 9 is:
Explanation: Sum of digits = 5 + 7 + y + 4 = 16 + y. For divisibility by 9, $16 + y$ must be a multiple of 9. The next multiple is 18, so y = 2.
15. 53724 is divisible by 11 because:
Explanation: Using the divisibility rule of 11: $5 - 3 + 7 - 2 + 4 = 11$. Since 11 is divisible by 11, the number is divisible by 11.
16. The smallest digit that can replace a in 8a36 so that the number is divisible by 11 is:
Explanation: Apply the rule of 11: $8 - a + 3 - 6 = 5 - a$. This must be 0 or a multiple of 11. For the smallest digit, take $5-a=0$, so $a=5$.
17. A number is divisible by 6 if it is divisible by:
Explanation: Since 6 = 2 x 3, a number must be divisible by both 2 and 3 to be divisible by 6.
18. Which of the following numbers is divisible by 5?
Explanation: A number is divisible by 5 if its last digit is 0 or 5. Among the options, only 980 ends in 0.
19. How many digits can replace x in 62x4 so that the number is divisible by 3?
Explanation: Digit sum = 6 + 2 + x + 4 = 12 + x. Since 12 is divisible by 3, x must also be divisible by 3. Possible digits: 0, 3, 6, 9. Total = 4 digits.
20. The divisibility rule of 11 works because, in modulo 11 arithmetic, 10 is congruent to:
Explanation: The rule of 11 comes from the fact that $10 \equiv -1 \pmod{11}$. So powers of 10 alternate as +1, -1, +1, -1, which creates the alternating sum rule.
21. The prime factorisation of 180 is:
Explanation: $180 = 2 x 90 = 2^2 x 45 = 2^2 x 3^2 x 5$.
22. How many positive factors does 72 have?
Explanation: $72 = 2^3 x 3^2$. Number of factors = $(3+1)(2+1) = 4 x 3 = 12$.
23. How many positive factors does 144 have?
Explanation: $144 = 2^4 x 3^2$. Number of factors = $(4+1)(2+1) = 5 x 3 = 15$.
24. Which of the following is a factor of 48?
Explanation: A factor divides the number exactly. $48 / 12 = 4$, so 12 is a factor of 48.
25. The highest power of 2 that exactly divides 96 is:
Explanation: $96 = 2^5 x 3$. So the highest power of 2 that divides 96 is $2^5 = 32$.
26. How many numbers between 1 and 50 are divisible by 6?
Explanation: Count multiples of 6 up to 50: $\lfloor 50/6 \rfloor = 8$.
27. Which of the following is a prime number?
Explanation: Check divisibility of 61 by primes up to $\sqrt{61}$, which is less than 8. It is not divisible by 2, 3, 5, or 7, so 61 is prime.
28. The number of positive factors of a perfect square is always:
Explanation: Factors usually come in pairs, like $a$ and $n/a$. For a perfect square, one factor repeats in the middle, namely $\sqrt{n}$, so the total becomes odd.
29. The product of two consecutive integers is 156. The integers are:
Explanation: Since $\sqrt{156}$ is close to 12.5, try 12 and 13. Indeed, $12 x 13 = 156$.
30. Seat numbers from 1 to 120 are printed in a hall in Mumbai. How many of them are divisible by both 3 and 5?
Explanation: A number divisible by both 3 and 5 must be divisible by 15. $\lfloor 120/15 \rfloor = 8$.
31. What is the remainder when 53 is divided by 7?
Explanation: $53 = 7 x 7 + 4$. So the remainder is 4.
32. What is the remainder when 248 is divided by 9?
Explanation: Since $10 \equiv 1 \pmod 9$, we can add digits. $2 + 4 + 8 = 14$, and $14 \equiv 5 \pmod 9$.
33. The remainder when 38 x 27 is divided by 5 is:
Explanation: $38 \equiv 3 \pmod 5$ and $27 \equiv 2 \pmod 5$. So $(38 x 27) \equiv 3 x 2 = 6 \equiv 1 \pmod 5$.
34. The remainder when 7^5 is divided by 6 is:
Explanation: $7 \equiv 1 \pmod 6$, so $7^5 \equiv 1^5 = 1 \pmod 6$.
35. The unit digit of 2^17 is:
Explanation: Unit digit cycle of powers of 2 is 2, 4, 8, 6. Cycle length = 4. Since 17 leaves remainder 1 on division by 4, take the first term, 2.
36. The unit digit of 3^26 is:
Explanation: Unit digit cycle of powers of 3 is 3, 9, 7, 1. Since 26 leaves remainder 2 on division by 4, the answer is the second term, 9.
37. The unit digit of 7^103 is:
Explanation: Unit digit cycle of powers of 7 is 7, 9, 3, 1. Since 103 leaves remainder 3 on division by 4, the answer is the third term, 3.
38. The unit digit of 9^48 is:
Explanation: Unit digit cycle of 9 is 9, 1. Cycle length = 2. Since 48 leaves remainder 0 on division by 2, take the last term of the cycle, 1.
39. What is the smallest number that must be added to 278 to make it divisible by 6?
Explanation: $278 = 6 x 46 + 2$. The remainder is 2, so to reach the next multiple of 6, add $6 - 2 = 4$.
40. What is the greatest number that must be subtracted from 278 to make it divisible by 6?
Explanation: $278 = 6 x 46 + 2$. The greatest number to subtract is the remainder itself, so subtract 2.