The product of four consecutive positive integers:
Here, four consecutive numbers mean if the first number is chosen at random, then the other three numbers which come one after another successively are chosen automatically.
(1) If 1 is chosen as the first number, then the other three consecutive numbers are 2, 3, and 4.
If the number 10 is chosen as the first number, then the other three consecutive numbers are 11, 12, and 13.
The product of numbers means they are being multiplied. AxBxCxD= DxCxBxA = C; C is the product/result of this multiplication.
Interchanging the place of numbers, AxBxCxD= DxCxBxA does not change the product, C, because the multiplication of whole numbers possesses the commutative property.
Positive Integers are 1, 2, 3, 4, 5…Z+
Odd numbers are not divisible by 2.
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21 etc
Prime numbers are either divisible by 1 or by the number itself.
Prime numbers from 1 to 100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
Every prime number, leaving 2 alone, is an odd number, but every odd number is not a prime number. 2 is the only prime number which is even. 9, 15, 21 etc. are odd numbers, but they are not prime numbers.
The number 1 is an odd number. Negative numbers are not prime numbers. The number 15 is an odd number, but it is not a prime number because it is divisible by 3 and 5 also.
When a number n is multiplied by n (by the number itself) its product becomes a perfect square.
n x n =n2
Ex. 2×2 =4; n=2
3×3= 9; n=3
4×4= 12; n=4
Here the product of the four consecutive positive integers is designated x. To which, 1 must be added to find N.
All three conditions will be checked for N, not for x which is the product of any four consecutive positive integers.
The best decision criterion in such problem is to prove that the product of any of the four consecutive integers is a prime number by taking four small numbers such as 1x2x3x4= 25. Once you prove that it is a prime number, this means it is also an odd number, but the opposite is not true.
1x2x3x4= x=24, x+1=25 it is an odd number and a perfect square of 5 but not a prime number
Here, we should stop checking the given conditions because at least in one case, we have found the second condition is wrong. That is sufficient to disapprove the second condition, and conclude that only the conditions A and C are correct; no matter in other cases the condition B turns out to be correct.
(Because the question is saying for all n=x+1)
However, to understand this type of question perfectly, we are expanding the solution.
2x3x4x5= x=120, x+1=121; it is an odd number and a perfect square of 11. But it is also a prime number
3x4x5x6=x=360. x+1= 361 it is an odd number and a perfect square of 11.
But it is also a prime number.
Checking Condition—- B. n is prime or not
To identify a prime number when N is greater than 100,
Firstly find out the square root of that number (N), and
Then, find the prime numbers coming before the square root of that number.
If the given number, N, is divisible by any of the prime numbers coming before its square root, then it (N) is not a prime number and if not divisible; then it is a prime number.
The square root of 361 is 19 and the prime numbers coming before it are:
2, 3, 5, 7, 11, 13, 17
Since 361(N) is not divisible by any of the prime numbers listed above. It is a prime number.
Mostly, a prime number never has a unit digit of 0, 2, 4, 5, 6 or 8 and the sum of the digits of that number never becomes the multiple of 3.
Hence, all three conditions are true in this case.
For example if we choose 10x11x12x13= 17,160
This means x=17160, and add 1 to it
It is a perfect square of 131.
The sum of the digit is 1+7+1+6+1=16 which is not a multiple of 3. Hence it is a prime number, and all the three conditions are true.
In all the four examples above:
At least N was always an odd number.
At least N was always a perfect square.
But it was not a prime number in case of the first four positive integers i.e. 1x2x3x4, but it turned out to be a prime number in case of four positive consecutive integers starting from 2x3x4x5.
Only the conditions (A) and (C) are found to be correct in all cases but not the condition (B).
Hence the option (a) is correct.
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