Match List I with List II
| List I | List II | ||
|---|---|---|---|
| A. | LCM of \(2^2\times3^3\times5^4\times7\) and \(2^3\times3\times5\times7^4\) | I. | \(2^6\times7^4\) |
| B. | HCF of \(2^3\times3^2\times5\) and \(3^2\times2^4\times5^3\) | II. | \(2^{12}\times3^5\times5^7\times7^{14}\) |
| C. | LCM of \(5^7\times7^5\times2^{12}\) and \(3^5\times7^{14}\times2^6\) | III. | \(2^3\times3^3\times5^4\times7^4\) |
| D. | HCF of \(5^7\times7^5\times2^{12}\) and \(3^5\times7^4\times2^6\times11\) | IV. | \(2^3\times3^2\times5\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A-III, B-IV, C-II, D-I
With numbers written as products of prime powers, the LCM takes every prime with its highest power and the HCF takes only the common primes with their lowest power.
A. LCM of \(2^2\times3^3\times5^4\times7\) and \(2^3\times3\times5\times7^4\) — this matches III. \(2^3\times3^3\times5^4\times7^4\). The highest powers are \(2^3\), \(3^3\), \(5^4\) and \(7^4\).
B. HCF of \(2^3\times3^2\times5\) and \(3^2\times2^4\times5^3\) — this matches IV. \(2^3\times3^2\times5\). The lowest powers of the common primes are \(2^3\), \(3^2\) and \(5^1\).
C. LCM of \(5^7\times7^5\times2^{12}\) and \(3^5\times7^{14}\times2^6\) — this matches II. \(2^{12}\times3^5\times5^7\times7^{14}\). The highest powers are \(2^{12}\), \(3^5\), \(5^7\) and \(7^{14}\).
D. HCF of \(5^7\times7^5\times2^{12}\) and \(3^5\times7^4\times2^6\times11\) — this matches I. \(2^6\times7^4\). Only 2 and 7 are common, with lowest powers \(2^6\) and \(7^4\); 3, 5 and 11 are not in both numbers.