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Prove that the product of any 3 consecutive positive integer is divisible by 6.?
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Prove that the product of any 3 consecutive positive integer is divisi...
Let us three consecutive integers be, n, n + 1 and n + 2.
Whenever a number is divided by 3 the remainder obtained is either 0 or 1 or 2.
let n = 3p or 3p + 1 or 3p + 2, where p is some integer.
If n = 3p, then n is divisible by 3.
If n = 3p + 1, then n + 2 = 3p + 1 + 2 = 3p + 3 = 3(p + 1) is divisible by 3.
If n = 3p + 2, then n + 1 = 3p + 2 + 1 = 3p + 3 = 3(p + 1) is divisible by 3.
So that n, n + 1 and n + 2 is always divisible by 3.
⇒ n (n + 1) (n + 2) is divisible by 3.
Similarly, whenever a number is divided 2 we will get the remainder is 0 or 1.
∴ n = 2q or 2q + 1, where q is some integer.
If n = 2q, then n and n + 2 = 2q + 2 = 2(q + 1) are divisible by 2.
If n = 2q + 1, then n + 1 = 2q + 1 + 1 = 2q + 2 = 2 (q + 1) is divisible by 2.
So that n, n + 1 and n + 2 is always divisible by 2.
⇒ n (n + 1) (n + 2) is divisible by 2.
But n (n + 1) (n + 2) is divisible by 2 and 3.
∴ n (n + 1) (n + 2) is divisible by 6.
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Prove that the product of any 3 consecutive positive integer is divisi...
Proof that the product of any 3 consecutive positive integers is divisible by 6
Introduction:
When we multiply any three consecutive positive integers, at least one of them will be divisible by 2 and another by 3. This is because every third integer is divisible by 3, and every second integer is divisible by 2. Therefore, the product of any three consecutive positive integers will be divisible by 6.
Explanation:
Let's consider three consecutive positive integers: n, n+1, and n+2.
Proof:
- Case 1: If n is even, then n+1 is odd and n+2 is even.
- Case 2: If n is divisible by 3, then n+1 and n+2 are not divisible by 3.
- Case 3: If n is not divisible by 2 or 3, then n+1 is even and n+2 is divisible by 3.
In each case, at least one of the three consecutive numbers is divisible by 2 and one by 3. Thus, the product of n, n+1, and n+2 will be divisible by 2 * 3 = 6.
Therefore, we have shown that the product of any three consecutive positive integers is divisible by 6.
Introduction:
When we multiply any three consecutive positive integers, at least one of them will be divisible by 2 and another by 3. This is because every third integer is divisible by 3, and every second integer is divisible by 2. Therefore, the product of any three consecutive positive integers will be divisible by 6.
Explanation:
Let's consider three consecutive positive integers: n, n+1, and n+2.
Proof:
- Case 1: If n is even, then n+1 is odd and n+2 is even.
- Case 2: If n is divisible by 3, then n+1 and n+2 are not divisible by 3.
- Case 3: If n is not divisible by 2 or 3, then n+1 is even and n+2 is divisible by 3.
In each case, at least one of the three consecutive numbers is divisible by 2 and one by 3. Thus, the product of n, n+1, and n+2 will be divisible by 2 * 3 = 6.
Therefore, we have shown that the product of any three consecutive positive integers is divisible by 6.
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Prove that the product of any 3 consecutive positive integer is divisible by 6.?
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Prove that the product of any 3 consecutive positive integer is divisible by 6.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Prove that the product of any 3 consecutive positive integer is divisible by 6.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that the product of any 3 consecutive positive integer is divisible by 6.?.
Prove that the product of any 3 consecutive positive integer is divisible by 6.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Prove that the product of any 3 consecutive positive integer is divisible by 6.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove that the product of any 3 consecutive positive integer is divisible by 6.?.
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