Question

If \(653 x y\) is divisible by 80 then the value of \(x+y\) is : (a) 2 (b) 3 (c) 4 (d) 6

Short answer

Answer: (c) 4

Step-by-step solution

Identify the prime factors of the given number

First, we need to identify the prime factors of 80 so that we can determine the required factors for 653xy. We know that 80 = 2^4 × 5^1.

Analyze the prime factors of 653

Next, we'll find the prime factors of 653. Since 653 is a prime number, its only factors are itself and 1.

Determine the necessary factors for x and y

In order for 653xy to be divisible by 80, it must include the prime factors found in Step 1. This means that 653xy = 653 × 2^a × 5^b (where a is the power of 2, and b is the power of 5). For the least value of x+y, we know that: 1. There should be three factors of 2, as 80 has 2^4 in its prime factors. 2. The least value of a+b = x+y should be taken. We have two cases: Case 1: x-y = 0 Here, we can assign the factors as follows: x: 2^3 (three factors of 2) y: 5^1 (one factor of 5) x+y = 3+1 = 4 Case 2: x-y > 0 This means that either x or y has more factors than the other. Since we need the least value for x+y, this case will result in a higher value than in Case 1. Therefore, we do not need to consider it further.

Choose the correct answer

The least possible value for x + y will be 4. So, the correct answer is: (c) 4

Key concepts

Prime Factorization

Prime factorization is the process of expressing a number as a product of its prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. Understanding prime factorization helps in breaking down bigger numbers into smaller prime pieces, which is especially useful when determining divisibility.
In the original exercise, we determined the prime factors of 80 to understand what factors 653xy must include to be divisible by 80. By decomposing 80 into its prime components, we find:

• 80 = 2^4 × 5^1

This means any number divisible by 80 must also have at least four 2's and one 5 as factors. Understanding this process helps in solving problems related to divisibility and finding common factors between different numbers.

Least Common Multiple

The Least Common Multiple (LCM) of two or more integers is the smallest positive integer that is divisible by each of the numbers. While the exercise focused more on divisibility, understanding LCM is crucial when dealing with similar problems and finding efficient solutions in mathematics.
For example, if you're tasked to find a number that two other numbers divide evenly, the LCM provides that efficient solution. It involves taking the highest powers of all prime numbers appearing in the factorization of these numbers.

In the exercise, although we focused on prime factors needed for divisibility by 80, knowing the LCM helps in understanding why we focus on collecting these prime components to meet or exceed the requirements of divisibility.

Number Properties

Number properties describe how numbers interact with one another, including how they can be broken down, combined, or manipulated within operations. These properties form the building blocks for virtually all mathematical problem-solving. Some basic number properties include divisibility, even and odd numbers, and integer properties, which were subtly used in the original exercise.
In our exercise, we dealt with divisibility, reflecting how certain integers can divide another integer without leaving a remainder. Specifically, the problem revolved around finding the combination of numbers that, when multiplied with 653, allow 653xy to be divisible by 80.

• 653 must be combined with sufficient 2's and 5's (from its factors) to meet or exceed those in 80.
• This ensures that no remainder is left when 653xy is divided by 80, emphasizing the concept of divisibility.

The art of judging these number properties helps learners identify essential factors, enhancing their overall understanding of problem-solving in mathematics.