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Chapter 6: Problem 467

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

Short Answer

Expert verified
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

01

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.
02

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.
03

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.
04

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.
05

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

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