Question

The product of three consecutive natural numbers is divisible by (a) 6 (b) 5 (c) 7 (d) 4

Short answer

The product of three consecutive natural numbers is divisible by 6 (Option a). This is because the product of two consecutive numbers is divisible by both 2 and 3, and their least common multiple is 6. The other options (5, 7, and 4) do not guarantee divisibility for any three consecutive numbers.

Step-by-step solution

Analyze the product of two consecutive natural numbers

Let's first look at the product of two consecutive natural numbers, \(n(n + 1)\). Since one of these numbers is even (either \(n\) or \(n+1\)), their product is guaranteed to be even (divisible by 2). For a product to be divisible by 3, either one or both of the consecutive numbers have to be divisible by 3. For consecutive numbers, at least one of them will be divisible by 3 every three numbers (i.e., 0, 3, 6, 9, etc.). Therefore, their product is guaranteed to be divisible by 3.

Verify if Option (a) is correct

Option (a) states that the product of three consecutive numbers is divisible by 6. We determined in Step 1 that the product of consecutive numbers is divisible by both 2 and 3. Hence, their product should also be divisible by their least common multiple (LCM). The LCM of 2 and 3 is 6. Therefore, the product of consecutive natural numbers is indeed divisible by 6. Option (a) is correct.

Test Option (b)

Option (b) claims that the product of three consecutive natural numbers is divisible by 5. If a number is divisible by 5, its last digit must be either 0 or 5. However, there is no guarantee that one of the three consecutive numbers we chose will have a unit digit of 0 or 5. Thus, we can't confirm the divisibility by 5 for any three consecutive numbers. Option (b) is incorrect.

Test Option (c)

Option (c) states that the product of three consecutive natural numbers is divisible by 7. Again, there is no guarantee that one of the three consecutive numbers we chose will be divisible by 7, because every seventh number is divisible by 7. Option (c) is incorrect.

Test Option (d)

Option (d) claims that the product of three consecutive natural numbers is divisible by 4. We determined in Step 1 that the product of consecutive numbers is divisible by 2, but every fourth number is divisible by 4. Therefore, we can't confirm that the product of any three consecutive numbers will be divisible by 4. Option (d) is incorrect. In conclusion, the product of three consecutive natural numbers is divisible by: (a) 6 - Correct (b) 5 - Incorrect (c) 7 - Incorrect (d) 4 - Incorrect

Key concepts

Consecutive Natural Numbers

Consecutive natural numbers are numbers that follow each other in order, without any gaps. These numbers begin with 1 and move forward in succession, like 1, 2, 3, 4, and so on. In mathematics, the pattern of consecutive natural numbers can be represented as:

• If we start with a number \(n\), the consecutive numbers would be \(n\), \(n+1\), \(n+2\), etc.

This sequence is simple, yet it forms the basis for some very interesting mathematical properties!

When dealing with consecutive natural numbers, you may observe a repeating trait regarding their divisibility. Particularly, when you compile a product of two or more consecutive natural numbers, it tends to have predictable divisibility outcomes. For example, in any set of three consecutive numbers, every three numbers will always contain a multiple of 3. Similarly, among any group of doubles, like \(n\) and \(n+1\), one will always have a factor of 2 since every other number is even. These inherent characteristics are the key to solving many number-related problems in mathematics.

Least Common Multiple (LCM)

The Least Common Multiple, often abbreviated as LCM, is a crucial concept in mathematics, especially when working with multiples and divisibility rules. The LCM of two or more numbers is the smallest common multiple shared between them. In other words, it's the smallest number that all the given numbers can divide without leaving a remainder.

• To find the LCM of two numbers, identify the prime factors of each number.
• Next, take the highest power of each prime factor that appears in the factorization of any of the numbers.
• Multiply these together to get the LCM.

For example, for 2 and 3, their prime factors are quite straightforward since both are prime numbers themselves. The LCM is simply \[ 2 \times 3 = 6 \]This knowledge proves extremely useful when solving problems involving consecutive numbers, such as determining the divisibility of a product of consecutive numbers. In the case of the exercise above, knowing that the LCM of 2 and 3 is 6 allows us to conclude that the product of any three consecutive numbers, which includes multiples of both 2 and 3, will indeed be divisible by 6.

Number Theory

Number theory is a branch of pure mathematics devoted to the study of integers and integer-related problems. It dives into concepts such as prime numbers, divisibility rules, and properties of numbers.

Within number theory, divisibility rules are indispensable. These rules allow us to quickly determine if one number is divisible by another without performing full division. For instance:

• A number is divisible by 2 if it ends in an even number (0, 2, 4, 6, 8).
• A number is divisible by 3 if the sum of its digits is divisible by 3.
• A number is divisible by 5 if it ends in 0 or 5.

By applying these rules, we unlock various numerical patterns and solve intriguing mathematical problems more efficiently.

Number theory's significance can't be overstated, as it provides a strong foundation for mathematical reasoning. It's especially relevant in the context of this exercise where understanding the nature of consecutive numbers and their product helps in applying divisibility rules to reach a solution easily. Its principles enable us to explore and solve problems surrounding natural numbers in a structured and methodical way.