26 If (m=3^2 \times 5), what is the value of (m)?
Answer and explanation
Correct answer: C. 45
Explanation: Step 1: (3^2=9). Step 2: (9 \times 5=45), so (m=45). Step 3: Keeping the order of powers and multiplication clear reduces mistakes.
Mathematics
Prime Factorisation
अभाज्य गुणनखंडन
In Class 10 Mathematics, under the Real Numbers chapter, Prime Factorisation teaches students to express a composite number as a product of prime numbers. Students practise identifying prime factors and writing factorisations using multiplication and exponents. This foundational skill supports the Fundamental Theorem of Arithmetic and later work with HCF and LCM.
Practice questions
27 Which is the prime factorisation of (216)?
Answer and explanation
Correct answer: B. (2^3 \times 3^3)
Explanation: Step 1: (216=8 \times 27). Step 2: (8=2^3) and (27=3^3), so (216=2^3 \times 3^3). Step 3: Recognising cube numbers is very useful in medium-level questions.
28 Which number has prime factorisation (3^3 \times 7)?
Answer and explanation
Correct answer: B. 189
Explanation: Step 1: (3^3=27). Step 2: (27 \times 7=189), so the number is (189). Step 3: Remembering small powers helps you calculate faster.
29 Which form is correct for the prime factorisation of (540)?
Answer and explanation
Correct answer: A. (2^2 \times 3^3 \times 5)
Explanation: Step 1: Write (540=54 \times 10). Step 2: (54=2 \times 3^3) and (10=2 \times 5), so (540=2^2 \times 3^3 \times 5). Step 3: Splitting a large number into easy parts is a safe method.
30 If a number has prime factorisation (2^4 \times 3^2), by which number must it be divisible?
Answer and explanation
Correct answer: C. 72
Explanation: Step 1: A divisor must not need prime exponents greater than those available. Step 2: (72=2^3 \times 3^2), which is fully present in (2^4 \times 3^2). Step 3: For divisibility, match the exponent of each prime separately.
31 How many trailing zeros will (2^3 \times 5^2) have?
Answer and explanation
Correct answer: B. 2
Explanation: Step 1: A trailing zero is formed by a pair (10=2 \times 5). Step 2: The exponent of (2) is (3) and of (5) is (2), so (2) pairs can be formed. Step 3: For trailing zeros, take the smaller exponent of (2) and (5).
32 What is the prime factorisation of (120)?
Answer and explanation
Correct answer: A. (2^3 \times 3 \times 5)
Explanation: Step 1: Write (120=12 \times 10). Step 2: (12=2^2 \times 3) and (10=2 \times 5), so (120=2^3 \times 3 \times 5). Step 3: Do not forget to combine repeated prime factors from different parts.
33 What is the greatest odd factor of (2^5 \times 3)?
Answer and explanation
Correct answer: A. 3
Explanation: Step 1: An odd factor must not contain (2). Step 2: Removing (2^5) leaves only (3), so the greatest odd factor is (3). Step 3: For the greatest odd factor, remove all powers of (2).
34 By the least number should (2^2 \times 3^2) be multiplied to make it a perfect cube?
Answer and explanation
Correct answer: C. 6
Explanation: Step 1: In a perfect cube, each prime exponent must be a multiple of (3). Step 2: To make (2^2) into (2^3) and (3^2) into (3^3), multiply by (2 \times 3=6). Step 3: For a cube, exponents should be like (3,6,9).
35 By the least number should (2^3 \times 3) be multiplied to make it a perfect square?
Answer and explanation
Correct answer: C. 6
Explanation: Step 1: In a perfect square, every prime exponent must be even. Step 2: To make (2^3) into (2^4) and (3) into (3^2), multiply by (2 \times 3=6). Step 3: For a square, increase odd exponents by one to make them even.
36 What is the prime factorisation of (64)?
Answer and explanation
Correct answer: C. (2^6)
Explanation: Step 1: Divide (64) repeatedly by (2). Step 2: (64=2 \times 2 \times 2 \times 2 \times 2 \times 2=2^6). Step 3: (4^3) gives the value, but it is not prime factorisation because (4) is not prime.
37 Which statement about prime factorisation is correct?
Answer and explanation
Correct answer: B. Only prime numbers are bases in prime factorisation
Explanation: Step 1: Prime factorisation means writing a number as a product of prime numbers. Step 2: Therefore, the bases must be prime numbers only. Step 3: Treating (1) as a prime factor is a major mistake.
38 How many positive factors does (2^2 \times 3 \times 7) have?
Answer and explanation
Correct answer: C. 12
Explanation: Step 1: Add (1) to each exponent for total factors. Step 2: ((2+1)(1+1)(1+1)=3 \times 2 \times 2=12). Step 3: Do not forget primes with exponent (1).
39 If (360=2^a \times 3^b \times 5), what are the values of (a) and (b)?
Answer and explanation
Correct answer: B. (a=3, b=2)
Explanation: Step 1: Write (360=36 \times 10). Step 2: (36=2^2 \times 3^2) and (10=2 \times 5), so (360=2^3 \times 3^2 \times 5). Hence (a=3, b=2). Step 3: Add exponents of repeated prime factors.
40 What is the smallest prime factor of (2^3 \times 3^2)?
Answer and explanation
Correct answer: A. 2
Explanation: Step 1: The bases in prime factorisation are the prime factors. Step 2: Here the bases are (2) and (3), and the smallest is (2). Step 3: To find the smallest prime factor, do not focus on exponents.
41 Which is the prime factorisation of (75)?
Answer and explanation
Correct answer: B. (3 \times 5^2)
Explanation: Step 1: Write (75=3 \times 25). Step 2: Since (25=5^2), (75=3 \times 5^2). Step 3: Do not leave (25) in the final answer because it is not prime.
42 How many maximum times can (2^4 \times 5) be completely divided by (10)?
Answer and explanation
Correct answer: A. 1
Explanation: Step 1: (10=2 \times 5). Step 2: The exponent of (2) is (4) and of (5) is (1), so only (1) pair of (10) can be formed. Step 3: The maximum number of divisions by (10) is decided by the smaller exponent.
43 If (n=2^2 \times 3^2 \times 5^2), what type of number is (n)?
Answer and explanation
Correct answer: B. Perfect square
Explanation: Step 1: In a perfect square, all prime exponents are even. Step 2: Here every exponent is (2), so (n) is a perfect square. Step 3: To identify a perfect square, check whether exponents are even.
44 If (n=2^3 \times 3^3), what type of number is (n)?
Answer and explanation
Correct answer: B. Perfect cube
Explanation: Step 1: In a perfect cube, all prime exponents are multiples of (3). Step 2: Both exponents are (3), so (n) is a perfect cube. Step 3: For a perfect cube, check exponents using (3).
45 What is the greatest prime factor of (2^3 \times 3 \times 5)?
Answer and explanation
Correct answer: C. 5
Explanation: Step 1: The prime factors are the base numbers. Step 2: Here (2,3,5) are prime factors, and the greatest is (5). Step 3: When the greatest prime factor is asked, do not choose a composite number.
46 How many factors of (2^2 \times 3^3) will be divisible by (3)?
Answer and explanation
Correct answer: B. 9
Explanation: Step 1: A factor divisible by (3) must have exponent of (3) at least (1). Step 2: The exponent of (2) has (3) choices (0,1,2), and exponent of (3) has (3) choices (1,2,3). Total (3 \times 3=9). Step 3: For conditional factors, adjust exponent choices carefully.
47 How many factors of (2^4 \times 5^2) will be divisible by (10)?
Answer and explanation
Correct answer: B. 8
Explanation: Step 1: A factor divisible by (10) must contain both (2) and (5). Step 2: The exponent of (2) can be (1) to (4), giving (4) choices, and the exponent of (5) can be (1) to (2), giving (2) choices. Total (4 \times 2=8). Step 3: Divisibility by (10) needs both prime factors.
48 How many factors of (2^3 \times 3^2 \times 5) are perfect squares?
Answer and explanation
Correct answer: A. 4
Explanation: Step 1: In a square factor, every prime exponent must be even. Step 2: For (2), choices are (0,2), so (2) choices; for (3), choices are (0,2), so (2) choices; for (5), only (0), so (1) choice. Total (2 \times 2 \times 1=4). Step 3: Count even exponent choices separately.
49 Which property tells the uniqueness of prime factorisation?
Answer and explanation
Correct answer: B. Fundamental theorem of arithmetic
Explanation: Step 1: Every integer greater than (1) has a unique prime factorisation apart from the order. Step 2: This comes from the fundamental theorem of arithmetic. Step 3: Remember uniqueness of prime factorisation with this theorem.
50 Which statement about the prime factorisation of (1) is correct?
Answer and explanation
Correct answer: A. (1) has no prime factor
Explanation: Step 1: A prime number has exactly two positive factors. Step 2: (1) has only one positive factor, so it is not prime and has no prime factor. Step 3: Do not make the mistake of treating (1) as prime.