The GCD of two numbers is 19 and their LCM is 285. One of the numbers is 57. Find the other number.

Question

31-10-2026
Mathematics

Answered

The GCD of two numbers is 19 and their LCM is 285. One of the numbers is 57. Find the other number.

Verified Answer :

Related Questions

The pedals on a bicycle give a mechanical advantage by allowing you to turn the pedals a distance to turn the circumstance of the wheels A long larger B long sh

For health care workers on the job the main risk of HIV transmission is froma being stuck with an HIVcontaminated needle or other sharp objectb coming into phys

Which equation has a graph that is perpendicular to the graph of 4x2y1

A line is a collection of

what led to the European era of exploration

Which of these is an illuminated source of light A the sun B a star C the earth seen from the moon D a meteorite as it burns up in the atmosphere ooh i dont kno

Qué significa la frase Al César lo que es del César y a Dios lo que es deDios Cómo debemos vivirla

The height of a triangle is 4 in greater than twice its base The area of the triangle is no more than 168 in2 Which inequality can be used to find the possible

Mi hermano y yo llegamos a la casa de padres nuestra nuestros sus tu

hoyts theory of urban land use and development emerged in response to changes in A agriculture B transportation C manufacturing D communication

GinnyAnswer GinnyAnswer · Answer 1 · 2026-10-31T12:39:33-04:00

Identify the relationship between two numbers, their GCD, and their LCM: \(A \times B = \gcd(A, B) \times \text{lcm}(A, B)\).
Substitute the given values \(A = 57\), \(\gcd = 19\), and \(\text{lcm} = 285\) into the formula: \(57 \times B = 19 \times 285\).
Solve for \(B\): \(B = \frac{19 \times 285}{57} = \frac{285}{3} = 95\).
The final answer is \(\boxed{95}\).

Explanation

Identify the Fundamental Relationship

The problem provides the greatest common divisor (GCD) and the least common multiple (LCM) of two numbers, and one of those numbers, say \(A = 57\). We are asked to find the other number, \(B\). The key relationship between two numbers \(A\) and \(B\), their GCD, and their LCM is given by the formula:
\[A \times B = \gcd(A, B) \times \text{lcm}(A, B)\]
We are given:
\(\gcd(A, B) = 19\)
\(\text{lcm}(A, B) = 285\)
\(A = 57\)

Calculate the Second Number

Substitute the given values into the relationship formula:
\[57 \times B = 19 \times 285\]
To find \(B\), we isolate it by dividing both sides of the equation by \(57\):
\[B = \frac{19 \times 285}{57}\]
We can simplify this calculation by noting that \(57 = 3 \times 19\):
\[B = \frac{19 \times 285}{3 \times 19} = \frac{285}{3}\]
Performing the final division, we get:
\[B = 95\]

Verify and Conclude

We have found that the other number is \(95\). To verify, we can check the GCD and LCM of \(57\) and \(95\):

Prime factorization of \(57 = 3 \times 19\)
Prime factorization of \(95 = 5 \times 19\)
\(\gcd(57, 95) = 19\)
\(\text{lcm}(57, 95) = 3 \times 5 \times 19 = 15 \times 19 = 285\)

The values match the original data, confirming the result. Thus, the other number is \(\boxed{95}\).

Examples

In logistics, finding the greatest common divisor and least common multiple is essential for coordinating schedules. For example, if two delivery trucks are on different routes—one returning every 57 minutes and another every 95 minutes—understanding how these cycles interact helps managers schedule maintenance or rest periods so they occur simultaneously. By knowing the GCD is \(\text{gcd}(57, 95) = 19\) and the LCM is \(\text{lcm}(57, 95) = 285\), the manager can plan for the trucks to meet at the depot every 285 minutes, ensuring operational efficiency.