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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-

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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.
$$\mathrm{Let}\:\mathrm{K}\:\mathrm{be}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{all}\:\mathrm{factors} \\ $$$$\left(\mathrm{b}−\mathrm{a}\right)\:\left(\mathrm{not}\:\mathrm{necessarily}\:\mathrm{distinct}\right) \\ $$$$\mathrm{where}\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}\:\mathrm{are}\:\mathrm{integers}\:\mathrm{satisfying} \\ $$$$\mathrm{1}\leqslant\mathrm{a}\leqslant\mathrm{b}\leqslant\mathrm{10}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{greatest} \\ $$$$\mathrm{integer}\:\mathrm{n}\:\mathrm{such}\:\mathrm{that}\:\mathrm{2}^{\mathrm{n}} \:\mathrm{divides}\:\mathrm{K}. \\ $$$$ \\ $$
Commented by Rasheed.Sindhi last updated on 08/Sep/20
a=b⇒ b−a=0 is one factor of K.So K=0 (If I understand the question)
$$ \\ $$$${a}={b}\Rightarrow \\ $$$${b}−{a}=\mathrm{0}\:{is}\:{one}\:{factor}\:{of}\:{K}.{So} \\ $$$${K}=\mathrm{0} \\ $$$$\left({If}\:{I}\:{understand}\:{the}\:{question}\right) \\ $$
Commented by Aina Samuel Temidayo last updated on 08/Sep/20
We are to find n.
$$\mathrm{We}\:\mathrm{are}\:\mathrm{to}\:\mathrm{find}\:\mathrm{n}. \\ $$
Commented by Rasheed.Sindhi last updated on 08/Sep/20
If K=0,what′s the question of greatest n 2^n ∣K ∀ n∈{0,1,2,3,...}
$${If}\:{K}=\mathrm{0},{what}'{s}\:{the}\:{question}\:{of} \\ $$$${greatest}\:{n} \\ $$$$\:\:\:\mathrm{2}^{{n}} \mid{K}\:\:\forall\:{n}\in\left\{\mathrm{0},\mathrm{1},\mathrm{2},\mathrm{3},…\right\} \\ $$
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 1≤a<b≤10^(Is it so?) . Find the greatest integer n such that 2^n divides K.
$$\mathrm{Let}\:\mathrm{K}\:\mathrm{be}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{all}\:\mathrm{factors} \\ $$$$\left(\mathrm{b}−\mathrm{a}\right)\:\left(\mathrm{not}\:\mathrm{necessarily}\:\mathrm{distinct}\right) \\ $$$$\mathrm{where}\:\mathrm{a}\:\mathrm{and}\:\mathrm{b}\:\mathrm{are}\:\mathrm{integers}\:\mathrm{satisfying} \\ $$$$\overset{{Is}\:{it}\:{so}?} {\mathrm{1}\leqslant\mathrm{a}<\mathrm{b}\leqslant\mathrm{10}}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{greatest} \\ $$$$\mathrm{integer}\:\mathrm{n}\:\mathrm{such}\:\mathrm{that}\:\mathrm{2}^{\mathrm{n}} \:\mathrm{divides}\:\mathrm{K}. \\ $$$$ \\ $$
Commented by Aina Samuel Temidayo last updated on 08/Sep/20
No.
$$\mathrm{No}. \\ $$$$ \\ $$
Answered by 1549442205PVT last updated on 09/Sep/20
If b≠a then K=(10−1)(10−2)...(10−9).(9−1) (9−2)...(9−8)....(3−1)(3−2).(2−1) =1!2!.3!....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
$$\mathrm{If}\:\mathrm{b}\neq\mathrm{a}\:\mathrm{then} \\ $$$$\mathrm{K}=\left(\mathrm{10}−\mathrm{1}\right)\left(\mathrm{10}−\mathrm{2}\right)…\left(\mathrm{10}−\mathrm{9}\right).\left(\mathrm{9}−\mathrm{1}\right) \\ $$$$\left(\mathrm{9}−\mathrm{2}\right)…\left(\mathrm{9}−\mathrm{8}\right)….\left(\mathrm{3}−\mathrm{1}\right)\left(\mathrm{3}−\mathrm{2}\right).\left(\mathrm{2}−\mathrm{1}\right) \\ $$$$=\mathrm{1}!\mathrm{2}!.\mathrm{3}!….\mathrm{9}! \\ $$$$\:\mathrm{9}!\mathrm{contain}\:\mathrm{7}\:\mathrm{factors}\:\mathrm{2}\: \\ $$$$\mathrm{8}!\mathrm{contain}\:\mathrm{7}\:\mathrm{factors}\:\mathrm{2} \\ $$$$\mathrm{7}!\mathrm{contain}\:\mathrm{4}\:\mathrm{factors}\:\mathrm{2} \\ $$$$\mathrm{6}!\mathrm{contain}\:\mathrm{4}\:\mathrm{factors}\:\mathrm{2} \\ $$$$\mathrm{5}!\mathrm{contain}\:\mathrm{3}\:\mathrm{factors}\:\mathrm{2} \\ $$$$\mathrm{4}!\mathrm{contain}\:\mathrm{3}\:\mathrm{factors}\:\mathrm{2} \\ $$$$\mathrm{3}!\mathrm{contain}\:\mathrm{1}\:\mathrm{factors}\:\mathrm{2} \\ $$$$\mathrm{2}!\mathrm{contain}\:\mathrm{1}\:\mathrm{factors}\:\mathrm{2} \\ $$$$\mathrm{Hence},\mathrm{K}\:\mathrm{contain}\:\mathrm{factor}\:\mathrm{2}^{\mathrm{30}\:} .\mathrm{Therefore}, \\ $$$$\mathrm{greatest}\:\mathrm{integer}\:\:\mathrm{number}\:\mathrm{n}\:\mathrm{satisfying}\: \\ $$$$\:\mathrm{2}^{\boldsymbol{\mathrm{n}}} \mid\mathrm{K}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{n}}=\mathrm{30}\: \\ $$
Commented by Rasheed.Sindhi last updated on 09/Sep/20
But Miss Aina insist that 1≤a≤b≤10
$${But}\:{Miss}\:{Aina}\:{insist}\:{that} \\ $$$$\mathrm{1}\leqslant{a}\leqslant{b}\leqslant\mathrm{10} \\ $$
Commented by 1549442205PVT last updated on 10/Sep/20
If it is such as then Sir have done. The problem has no answer
$$\mathrm{If}\:\mathrm{it}\:\mathrm{is}\:\mathrm{such}\:\mathrm{as}\:\mathrm{then}\:\mathrm{Sir}\:\mathrm{have}\:\mathrm{done}. \\ $$$$\mathrm{The}\:\mathrm{problem}\:\mathrm{has}\:\mathrm{no}\:\mathrm{answer} \\ $$
Commented by Aina Samuel Temidayo last updated on 10/Sep/20
It has.
$$\mathrm{It}\:\mathrm{has}. \\ $$
Commented by 1549442205PVT last updated on 13/Sep/20
Answer of Sir Shindi:∄ greatest n
$$\mathrm{Answer}\:\mathrm{of}\:\mathrm{Sir}\:\mathrm{Shindi}:\nexists\:\mathrm{greatest}\:\mathrm{n} \\ $$

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