Let-K-be-the-product-of-all-factors-b-a-not-necessarily-distinct-where-a-and-b-are-integers-satisfying-1-a-b-10-Find-the-greatest-integer-n-such-that-2-n-divides-K-

Question Number 112535 by Aina Samuel Temidayo last updated on 08/Sep/20

Let K be the product of all factors

(b - a) (not necessarily distinct)

where a and b are integers satisfying

1 ⩽ a ⩽ b ⩽ 10 . Find the greatest

integer n such that 2^{n} divides K .

Commented by Rasheed.Sindhi last updated on 08/Sep/20

a = b \Rightarrow

b - a = 0 i s o n e f a c t o r o f K . S o

K = 0

(I f I u n d e r s t a n d t h e q u e s t i o n)

Commented by Aina Samuel Temidayo last updated on 08/Sep/20

We are to find n .

Commented by Rasheed.Sindhi last updated on 08/Sep/20

I f K = 0, {w h a t}^{'} s t h e q u e s t i o n o f

g r e a t e s t n

2^{n} ∣ K \forall n \in {0, 1, 2, 3, \dots}

Commented by Rasheed.Sindhi last updated on 08/Sep/20

Let K be the product of all factors

(b - a) (not necessarily distinct)

where a and b are integers satisfying

\overset{I s i t s o ?}{1 ⩽ a < b ⩽ 10} . Find the greatest

integer n such that 2^{n} divides K .

Commented by Aina Samuel Temidayo last updated on 08/Sep/20

No .

Answered by 1549442205PVT last updated on 09/Sep/20

If b \neq a then

K = (10 - 1) (10 - 2) \dots (10 - 9) . (9 - 1)

(9 - 2) \dots (9 - 8) \dots . (3 - 1) (3 - 2) . (2 - 1)

= 1! 2! . 3! \dots . 9!

9! contain 7 factors 2

8! contain 7 factors 2

7! contain 4 factors 2

6! contain 4 factors 2

5! contain 3 factors 2

4! contain 3 factors 2

3! contain 1 factors 2

2! contain 1 factors 2

Hence, K contain factor 2^{30} . Therefore,

greatest integer number n satisfying

2^{n} ∣ K is n = 30

Commented by Rasheed.Sindhi last updated on 09/Sep/20

B u t M i s s A i n a i n s i s t t h a t

1 ⩽ a ⩽ b ⩽ 10

Commented by 1549442205PVT last updated on 10/Sep/20

If it is such as then Sir have done .

The problem has no answer

Commented by Aina Samuel Temidayo last updated on 10/Sep/20

It has .

Commented by 1549442205PVT last updated on 13/Sep/20

Answer of Sir Shindi : ∄ greatest n