Find the HCF of 24 × 33 × 5 × 75 × 135, 23 × 36 × 53 × 79, and 24 × 54 × 73 ×…
2025
Find the HCF of 24 × 33 × 5 × 75 × 135, 23 × 36 × 53 × 79, and 24 × 54 × 73 × 134.
- A.
24 × 36 × 53 × 73
- B.
23 × 33 × 53 × 73 × 134
- C.
24 × 33 × 5 × 73
- D.
23 × 5 × 73
Show answer & explanation
Correct answer: D
Concept: When numbers are expressed in prime-factorised form, their HCF (Highest Common Factor) is the product of every prime factor that is common to ALL the numbers, each raised to the LOWEST power in which it occurs across those numbers. Any prime that is absent from even one of the numbers cannot appear in the HCF at all.
Application: The three numbers are 24 × 33 × 5 × 75 × 135, 23 × 36 × 53 × 79, and 24 × 54 × 73 × 134. Listing the exponent of each prime in every number:
Prime | Number 1 | Number 2 | Number 3 | Common to all three? | Lowest power |
|---|---|---|---|---|---|
2 | 4 | 3 | 4 | Yes | 3 |
3 | 3 | 6 | absent | No | — |
5 | 1 | 3 | 4 | Yes | 1 |
7 | 5 | 9 | 3 | Yes | 3 |
13 | 5 | absent | 4 | No | — |
Only 2, 5 and 7 occur in all three numbers (3 is missing from the third number, and 13 is missing from the second), so the HCF is 23 × 5 × 73.
Cross-check: 23 × 5 × 73 uses only exponents that are less than or equal to the corresponding exponent in every one of the three numbers, so it divides all three; and no larger common power is possible for 2, 5 or 7 because 23, 51 and 73 are each the smallest exponent seen for that prime (bounded by number 2 for the power of 2, by number 1 for the power of 5, and by number 3 for the power of 7).
Therefore, the HCF is 23 × 5 × 73.