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

$$\mathrm{Let}\mathrm{K}\mathrm{be}\mathrm{the}\mathrm{product}\mathrm{of}\mathrm{all}\mathrm{factors}$$$$(\mathrm{b}-\mathrm{a})\left(\mathrm{not}\mathrm{necessarily}\mathrm{distinct}\right)$$$$\mathrm{where}\mathrm{a}\mathrm{and}\mathrm{b}\mathrm{are}\mathrm{integers}\mathrm{satisfying}$$$$1⩽\mathrm{a}⩽\mathrm{b}⩽10.\mathrm{Find}\mathrm{the}\mathrm{greatest}$$$$\mathrm{integer}\mathrm{n}\mathrm{such}\mathrm{that}{2}^{\mathrm{n}}\mathrm{divides}\mathrm{K}.$$$$$$

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

$$$$$$a=b\Rightarrow$$$$b-a=0isonefactorofK.So$$$$K=0$$$$\left(IfIunderstandthequestion\right)$$

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

$$\mathrm{We}\mathrm{are}\mathrm{to}\mathrm{find}\mathrm{n}.$$

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

$$IfK=0,{what}^{\prime }sthequestionof$$$$greatestn$$$$2^n\mid K\mathrm{\forall }n\in \{0,1,2,3,...\}$$

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

$$\mathrm{Let}\mathrm{K}\mathrm{be}\mathrm{the}\mathrm{product}\mathrm{of}\mathrm{all}\mathrm{factors}$$$$(\mathrm{b}-\mathrm{a})\left(\mathrm{not}\mathrm{necessarily}\mathrm{distinct}\right)$$$$\mathrm{where}\mathrm{a}\mathrm{and}\mathrm{b}\mathrm{are}\mathrm{integers}\mathrm{satisfying}$$$$\stackrel{Isitso?}{1⩽\mathrm{a}<\mathrm{b}⩽10}.\mathrm{Find}\mathrm{the}\mathrm{greatest}$$$$\mathrm{integer}\mathrm{n}\mathrm{such}\mathrm{that}{2}^{\mathrm{n}}\mathrm{divides}\mathrm{K}.$$$$$$

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

$$\mathrm{No}.$$$$$$

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

$$\mathrm{If}\mathrm{b}\ne \mathrm{a}\mathrm{then}$$$$\mathrm{K}=(10-1)(10-2)...(10-9).(9-1)$$$$(9-2)...(9-8)....(3-1)(3-2).(2-1)$$$$=1!2!.3!....9!$$$$9!\mathrm{contain}7\mathrm{factors}2$$$$8!\mathrm{contain}7\mathrm{factors}2$$$$7!\mathrm{contain}4\mathrm{factors}2$$$$6!\mathrm{contain}4\mathrm{factors}2$$$$5!\mathrm{contain}3\mathrm{factors}2$$$$4!\mathrm{contain}3\mathrm{factors}2$$$$3!\mathrm{contain}1\mathrm{factors}2$$$$2!\mathrm{contain}1\mathrm{factors}2$$$$\mathrm{Hence},\mathrm{K}\mathrm{contain}\mathrm{factor}{2}^{30}.\mathrm{Therefore},$$$$\mathrm{greatest}\mathrm{integer}\mathrm{number}\mathrm{n}\mathrm{satisfying}$$$$2^{\mathbf{n}}\mid \mathrm{K}\mathbf{is}\mathbf{n}=30$$

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

$$ButMissAinainsistthat$$$$1⩽a⩽b⩽10$$

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

$$\mathrm{If}\mathrm{it}\mathrm{is}\mathrm{such}\mathrm{as}\mathrm{then}\mathrm{Sir}\mathrm{have}\mathrm{done}.$$$$\mathrm{The}\mathrm{problem}\mathrm{has}\mathrm{no}\mathrm{answer}$$

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

$$\mathrm{It}\mathrm{has}.$$

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

$$\mathrm{Answer}\mathrm{of}\mathrm{Sir}\mathrm{Shindi}:\nexists \mathrm{greatest}\mathrm{n}$$

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