Two numbers have these properties. - Both numbers are greater than 6 . - Their highest common factor (HCF) is 6 . - Their lowest common multiple (LCM) is 60 . Find the two numbers.

Let the two numbers be \( a \) and \( b \). Given the properties: 1. Both numbers are greater than 6. 2. The highest common factor (HCF) of \( a \) and \( b \) is 6. 3. The lowest common multiple (LCM) of \( a \) and \( b \) is 60. We can use the relationship between HCF and LCM, which states: \[ \text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b \] Substituting the known values: \[ 6 \times 60 = a \times b \] This simplifies to: \[ 360 = a \times b \] Since the HCF is 6, we can express \( a \) and \( b \) in terms of their HCF: \[ a = 6m \quad \text{and} \quad b = 6n \] where \( m \) and \( n \) are coprime integers (since the HCF of \( a \) and \( b \) is 6). Substituting these into the product equation gives: \[ (6m)(6n) = 360 \] This simplifies to: \[ 36mn = 360 \] Dividing both sides by 36: \[ mn = 10 \] Now we need to find pairs of coprime integers \( (m, n) \) such that their product is 10. The pairs of factors of 10 are: - \( (1, 10) \) - \( (2, 5) \) Now we check which pairs are coprime: 1. \( (1, 10) \) are coprime. 2. \( (2, 5) \) are coprime. Now we can find the corresponding values of \( a \) and \( b \): 1. For \( (m, n) = (1, 10) \): \[ a = 6 \times 1 = 6, \quad b = 6 \times 10 = 60 \] (Not valid since both numbers must be greater than 6.) 2. For \( (m, n) = (2, 5) \): \[ a = 6 \times 2 = 12, \quad b = 6 \times 5 = 30 \] (Both numbers are greater than 6.) Now we check the HCF and LCM: - HCF of 12 and 30: - The factors of 12 are \( 1, 2, 3, 4, 6, 12 \). - The factors of 30 are \( 1, 2, 3, 5, 6, 10, 15, 30 \). - The common factors are \( 1, 2, 3, 6 \), so \( \text{HCF}(12, 30) = 6 \). - LCM of 12 and 30: - The multiples of 12 are \( 12, 24, 36, 48, 60, \ldots \). - The multiples of 30 are \( 30, 60, 90, \ldots \). - The least common multiple is \( 60 \). Both conditions are satisfied. Therefore, the two numbers are: \[ \boxed{12} \quad \text{and} \quad \boxed{30} \]