Factorial division
( n − 1 ) ! ( n + 2 ) ! The above fraction is divisible by which of the following?
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4 solutions
I don't like the choices for the answers; none of them are incorrect. And answering with 6 immediately implies the rest.
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Typically in these MCQ-type questions, only one option is correct to the exclusion of all others Therefore, if someone chooses any option other than "All of these" they are assuming that the other options are incorrect, which is wrong. Therefore only "All of these" is the correct answer.
That's a great compliment for me !! . More the people get tricked , more it denotes the success of my problem .!!!!
( ⌣ ¨ )
They should remove 1 as an option as 0!=1 is not something the average person should be expected to just know. And maybe explain the ! symbol with a question like this.
The given fraction = (n - 1)! n (n + 1) (n +2)/(n - 1)! = n (n + 1) (n +2)
n (n + 1) (n +2) is the product of 3 consecutive integers, so it is divisible by 2 and 3
If the expression is divisible by 2 and 3,then it must be divisible by 6
Thank you! I wasn't understanding why it was divisible by 3.
Ah. Every product of 3 consecutive integers is divisible by 2 and 3. That's in part because at least one even number exists in that pattern. How can you prove mathematically that for any such product, it is divisible by 3, but without using a proof-by-induction?
Plug in n = 1. 6 is divisible by everything.
Proof by induction. Love it 👏🏾👏🏾👏🏾
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Is that allowed?
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Yes but this is just the initial case of the induction proof. So the proof is incomplete. Next you have to prove for n+1. It will work although it's more elaborate
Incomplete, it's just stating the initial case.
Firstly simplify:
( n + 2 ) ( n + 1 ) n ( n − 1 ) ! = ( n + 2 ) !
so
( n − 1 ) ! ( n + 2 ) ! = ( n − 1 ) ! ( n + 2 ) ( n + 1 ) n ( n − 1 ) ! ) = ( n + 2 ) ( n + 1 ) n
all numbers are divisible by 1. all even numbers are divisible by 2, so if n is even it will divide if n is odd then n+1 is even and it will still divide. so more than 1 answer is correct so looking at the options you pick a single answer or all of them it must be all of them by default.
Although you can work it out mathematically but why bother if you have the answer?
That's what I did, but you gotta assume that trick (where if two of them work then all of them work) isn't there.
( n − 1 ) ! ( n + 2 ) ! = ( n − 1 ) ! ( n − 1 ) ! × n × ( n + 1 ) × ( n + 2 ) = n ( n + 1 ) ( n + 2 ) Every second number is divisible by 2.So the expression is divisible by 2.
Every third number is divisible by 3.So the expression is divisible by 3.
If the expression is divisible by 2 and 3,then it must be divisible by 2 × 3 = 6
Every expression is divisible by 1.
So the answer is A l l O f T h e s e