The HCF of two numbers is 13 and the other two factors of their LCM are 5 and 6. The larger of two numbers is?

Steps to Solve

1. Identify HCF and LCM The HCF of the two numbers is given as 13, and the LCM can be calculated using the additional factors provided: 5 and 6.

2. Calculate LCM To find the LCM, we multiply the HCF by the other factors: $$ \text{LCM} = \text{HCF} \times 5 \times 6 $$ Substituting the values: $$ \text{LCM} = 13 \times 5 \times 6 = 390 $$

3. Use the relationship between HCF and LCM The relationship states that: $$ \text{HCF} \times \text{LCM} = \text{number 1} \times \text{number 2} $$ Let the two numbers be $a$ and $b$. Thus, $$ 13 \times 390 = a \times b $$ Calculating the product: $$ 13 \times 390 = 5070 $$

4. Express the numbers in terms of HCF Since both numbers are multiples of the HCF, we can write: $$ a = 13x \quad \text{and} \quad b = 13y $$ Substituting these into the equation: $$ 5070 = (13x)(13y) = 169xy $$ Simplifying gives us: $$ xy = \frac{5070}{169} = 30 $$

5. Find pairs of factors for 30 Now we find pairs of integers $(x,y)$ that satisfy $xy = 30$: The pairs (considering positive integers) are:

• (1, 30)
• (2, 15)
• (3, 10)
• (5, 6)

6. Calculate the larger number of each pair For each pair $(x,y)$, the corresponding numbers will be:

• For (1, 30): $13 \times 1 = 13$, $13 \times 30 = 390$
• For (2, 15): $13 \times 2 = 26$, $13 \times 15 = 195$
• For (3, 10): $13 \times 3 = 39$, $13 \times 10 = 130$
• For (5, 6): $13 \times 5 = 65$, $13 \times 6 = 78$

The larger number for each pair is:

• 390
• 195
• 130
• 78

7. Determine the maximum larger number The maximum value from our pairs is: $$ \text{Maximum larger number} = 390 $$