(Q). Consider a number x.(x + 1).(x + 2).(x + 3)………till x terms. If above number is completely divisible by all two digit prime numbers then what is the minimum value of x.
Solution:-
Since largest two digit prime number is = 97 & Since above number is completely divisible by all two digit prime numbers.
Then, let the highest term of the expression be 97 then expression ⇒
49×50×51×52×…………..till 49 terms.
= 49×50×51×52×………….×97
Hence every prime number from 49 till 97 will divide this expression completely.
& prime numbers which are less than 49 ⇒
Let 47
Then 47×2 = 94 ✔
Let 43
Then 43×2 = 86 ✔
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Let 2
Then 2×25 = 50 ✔
Let 13
Then 13×5 = 65 ✔
Hence when x = 49 then all two digits prime number divide this expression.
Hence minimum value of x = 49 Answer