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SubjectsMathematics
अभाज्य गुणनखंडन द्वारा महत्तम समापवर्तक और लघुत्तम समापवर्त्य
In this Class 10 Mathematics topic from Real Numbers, students learn to find the HCF and LCM of two or more numbers through prime factorisation. They break each number into prime factors, compare the resulting powers, and select the appropriate common factors: the lowest powers for HCF and the highest powers for LCM. The topic also develops accuracy in writing factor trees, simplifies comparison between methods, and helps students apply these concepts to numerical and everyday problem-solving situations.
Correct answer: A. (2^4\times3^2\times5)
Explanation: Step 1: HCF uses only common prime factors. Step 2: The common primes are (2), (3), and (5), with smaller powers (2^4), (3^2), and (5). Step 3: In exponent questions, choose the smaller power for each base separately.
Correct answer: A. (2^5\times3^4\times5\times7\times11)
Explanation: Step 1: LCM uses the highest power of every prime factor present. Step 2: The highest powers are (2^5), (3^4), (5), (7), and (11). Step 3: Include a prime even if it appears in only one number.
Correct answer: A. (324)
Explanation: Step 1: For two numbers, product equals HCF times LCM. Step 2: The other number is (\frac{36\times1620}{180}=324). Step 3: Write the relation first and then simplify the division.
Correct answer: A. (2^3\times3^2\times5\times7)
Explanation: Step 1: HCF is (2^2\times3^3\times7), and LCM is (2^5\times3^5\times5\times7^2). Step 2: On division, subtract powers to get (2^3\times3^2\times5\times7). Step 3: Exponent subtraction is very useful in ratio questions.
Correct answer: A. Such whole numbers are possible
Explanation: Step 1: For two whole numbers, the HCF must exactly divide the LCM. Step 2: (1320\div24=55), which is a whole number, so such numbers can exist. Step 3: For existence checks, test this divisibility condition first.
Correct answer: B. (4)
Explanation: Step 1: Let the numbers be (18m) and (18n), where (m) and (n) are coprime. Step 2: (18mn=1260), so (mn=70=2\times5\times7); this gives (4) unordered coprime factor pairs. Step 3: For a square-free product, split prime factors into two groups to count unordered pairs.
Correct answer: A. (293)
Explanation: Step 1: Subtracting (13) makes the number divisible by (42), (56), and (70). Step 2: Their LCM is (280), so the smallest number is (280+13=293). Step 3: In common-remainder questions, add the remainder at the end.
Correct answer: A. (15)
Explanation: Step 1: For the same remainder, take the HCF of the differences of the given numbers. Step 2: The differences are (165), (255), and (420); their HCF is (15). Step 3: In same-remainder divisor questions, work with differences.
Correct answer: A. (a=4), (b=3)
Explanation: Step 1: HCF uses the smaller power of each common prime. Step 2: The smaller power of (2) must be (4), so (a=4) is possible; the smaller power of (3) must be (3), so (b=3) is possible. Step 3: Check unknown powers separately for each prime base.
Correct answer: A. (a=6), (b=4)
Explanation: Step 1: LCM uses the highest power of each prime. Step 2: The highest power of (3) must be (6), so (a=6); the highest power of (5) must be (4), so (b=4). Step 3: In unknown-power LCM questions, identify the maximum power.
Correct answer: B. (66)
Explanation: Step 1: The maximum number of identical groups is found by HCF. Step 2: (198=2\times3^2\times11) and (330=2\times3\times5\times11), so HCF (=2\times3\times11=66). Step 3: Use HCF for maximum equal distribution.
Correct answer: A. (1260)
Explanation: Step 1: The next common ringing time is the LCM of the intervals. Step 2: (45=3^2\times5), (60=2^2\times3\times5), and (84=2^2\times3\times7), so LCM (=2^2\times3^2\times5\times7=1260). Step 3: Use LCM for repeated-time questions.
Correct answer: C. (7)
Explanation: Step 1: Product of two numbers equals HCF times LCM. Step 2: The powers of (3) are (2) and (5), so the total power is (7). Step 3: Exponents of the same base add during multiplication.
Correct answer: C. (6:1)
Explanation: Step 1: LCM (=\frac{154\times231}{77}=462). Step 2: (\frac{462}{77}=6), so the ratio is (6:1). Step 3: Always write ratios in simplest form.
Correct answer: B. (32)
Explanation: Step 1: The maximum number of equal parts is found by HCF. Step 2: (96=2^5\times3), (160=2^5\times5), and (224=2^5\times7), so HCF (=2^5=32). Step 3: In maximum equal division questions, identify HCF.
Correct answer: B. (72000)
Explanation: Step 1: The smallest number exactly divisible by all is the LCM. Step 2: (64=2^6), (72=2^3\times3^2), and (125=5^3), so LCM (=2^6\times3^2\times5^3=72000). Step 3: Keeping the highest powers correctly gives the right answer.
Correct answer: A. The HCF will be (A)
Explanation: Step 1: (B=A\times11), so (A) exactly divides (B). Step 2: When one number divides the other exactly, the smaller number is the HCF. Step 3: Identifying a multiple relation saves time.
Correct answer: B. (42)
Explanation: Step 1: After factoring out HCF (40), (r) and (s) are coprime. Step 2: LCM (=40rs=1680), so (rs=42). Step 3: Factor out the HCF to make the question shorter.
Correct answer: B. (2)
Explanation: Step 1: Compare the powers of (3). Step 2: (135=3^3\times5), (180=2^2\times3^2\times5), and (225=3^2\times5^2), so the smallest power is (2). Step 3: HCF uses the smallest power.
Correct answer: B. (3)
Explanation: Step 1: Prime factorise: (80=2^4\times5), (144=2^4\times3^2), and (225=3^2\times5^2). Step 2: The distinct primes in the LCM are (2), (3), and (5), so the count is (3). Step 3: Do not count powers as separate primes.
Correct answer: A. (3) and (1)
Explanation: Step 1: LCM takes the higher power, while HCF takes the lower power. Step 2: The powers of (5) are (1) and (3), so (L) has power (3) and (H) has power (1). Step 3: Do not interchange the two rules.
Correct answer: A. (1)
Explanation: Step 1: (121=11^2), (143=11\times13), and (169=13^2). Step 2: No prime factor is common to all three numbers, so the HCF is (1). Step 3: For three numbers, a common factor must appear in all of them.
Correct answer: C. (14)
Explanation: Step 1: (32=2^5) and (1536=2^9\times3). Step 2: Product equals HCF times LCM, so the power of (2) is (5+9=14). Step 3: Add exponents when multiplying powers with the same base.
Correct answer: B. (1224)
Explanation: Step 1: LCM (=\frac{180\times252}{36}=1260). Step 2: The difference is (1260-36=1224). Step 3: When difference is asked, find both values clearly first.
Correct answer: D. (899)
Explanation: Step 1: Coprime numbers have HCF (1). Step 2: Product (=) HCF (\times) LCM, so the LCM is (899). Step 3: For coprime numbers, the LCM equals the product.
