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Question Number 22059 Answers: 1 Comments: 0

A flywheel whose diameter is 1.5m decrease uniformly from 240rad/min until it came to rest 10s. Find the number of revolution made.

$$\mathrm{A}\:\mathrm{flywheel}\:\mathrm{whose}\:\mathrm{diameter}\:\mathrm{is}\:\mathrm{1}.\mathrm{5m}\:\mathrm{decrease}\:\mathrm{uniformly}\:\mathrm{from}\:\mathrm{240rad}/\mathrm{min} \\ $$$$\mathrm{until}\:\mathrm{it}\:\mathrm{came}\:\mathrm{to}\:\mathrm{rest}\:\mathrm{10s}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{revolution}\:\mathrm{made}. \\ $$

Question Number 22058 Answers: 1 Comments: 0

Two balls of mass 500g and 750g moving with 15m/s and 10m/s towards each other collides. Find the velocities of the ball after collision, if the coefficient of restitution is 0.8

$$\mathrm{Two}\:\mathrm{balls}\:\mathrm{of}\:\mathrm{mass}\:\mathrm{500g}\:\mathrm{and}\:\mathrm{750g}\:\mathrm{moving}\:\mathrm{with}\:\mathrm{15m}/\mathrm{s}\:\mathrm{and} \\ $$$$\mathrm{10m}/\mathrm{s}\:\mathrm{towards}\:\mathrm{each}\:\mathrm{other}\:\mathrm{collides}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{velocities}\:\mathrm{of}\:\mathrm{the}\:\mathrm{ball}\:\mathrm{after} \\ $$$$\mathrm{collision},\:\mathrm{if}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{restitution}\:\mathrm{is}\:\mathrm{0}.\mathrm{8} \\ $$

Question Number 22057 Answers: 0 Comments: 0

∫f(x)dx=((d/dx))^2 f(x)=???

$$\int{f}\left({x}\right){dx}=\left(\frac{{d}}{{dx}}\right)^{\mathrm{2}} \\ $$$${f}\left({x}\right)=??? \\ $$

Question Number 22055 Answers: 0 Comments: 1

Question Number 22052 Answers: 0 Comments: 2

A hockey player is moving northward and suddenly turns westward with the same speed to avoid an opponent. The force that acts on the player is (a) frictional force along westward (b) muscle force along southward (c) frictional force along south-west (d) muscle force along south-west

$$\mathrm{A}\:\mathrm{hockey}\:\mathrm{player}\:\mathrm{is}\:\mathrm{moving}\:\mathrm{northward} \\ $$$$\mathrm{and}\:\mathrm{suddenly}\:\mathrm{turns}\:\mathrm{westward}\:\mathrm{with} \\ $$$$\mathrm{the}\:\mathrm{same}\:\mathrm{speed}\:\mathrm{to}\:\mathrm{avoid}\:\mathrm{an}\:\mathrm{opponent}. \\ $$$$\mathrm{The}\:\mathrm{force}\:\mathrm{that}\:\mathrm{acts}\:\mathrm{on}\:\mathrm{the}\:\mathrm{player}\:\mathrm{is} \\ $$$$\left({a}\right)\:\mathrm{frictional}\:\mathrm{force}\:\mathrm{along}\:\mathrm{westward} \\ $$$$\left({b}\right)\:\mathrm{muscle}\:\mathrm{force}\:\mathrm{along}\:\mathrm{southward} \\ $$$$\left({c}\right)\:\mathrm{frictional}\:\mathrm{force}\:\mathrm{along}\:\mathrm{south}-\mathrm{west} \\ $$$$\left({d}\right)\:\mathrm{muscle}\:\mathrm{force}\:\mathrm{along}\:\mathrm{south}-\mathrm{west} \\ $$

Question Number 22047 Answers: 2 Comments: 1

If x > 0 and the 4^(th) term in the expansion of (2 + (3/8)x)^(10) has maximum value then find the range of x.

$$\mathrm{If}\:{x}\:>\:\mathrm{0}\:\mathrm{and}\:\mathrm{the}\:\mathrm{4}^{\mathrm{th}} \:\mathrm{term}\:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion} \\ $$$$\mathrm{of}\:\left(\mathrm{2}\:+\:\frac{\mathrm{3}}{\mathrm{8}}{x}\right)^{\mathrm{10}} \:\mathrm{has}\:\mathrm{maximum}\:\mathrm{value} \\ $$$$\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:{x}. \\ $$

Question Number 22050 Answers: 0 Comments: 0

Calculate the energy emitted when electrons of 1 g atom of hydrogen undergo transition giving the spectral line of lowest energy in the visible region of its atomic spectrum (R_H = 1.1 × 10^7 m^(−1) , c = 3 × 10^8 ms^(−1) , h = 6.62 × 10^(−34) Js)

$$\mathrm{Calculate}\:\mathrm{the}\:\mathrm{energy}\:\mathrm{emitted}\:\mathrm{when} \\ $$$$\mathrm{electrons}\:\mathrm{of}\:\mathrm{1}\:\mathrm{g}\:\mathrm{atom}\:\mathrm{of}\:\mathrm{hydrogen} \\ $$$$\mathrm{undergo}\:\mathrm{transition}\:\mathrm{giving}\:\mathrm{the}\:\mathrm{spectral} \\ $$$$\mathrm{line}\:\mathrm{of}\:\mathrm{lowest}\:\mathrm{energy}\:\mathrm{in}\:\mathrm{the}\:\mathrm{visible} \\ $$$$\mathrm{region}\:\mathrm{of}\:\mathrm{its}\:\mathrm{atomic}\:\mathrm{spectrum} \\ $$$$\left(\mathrm{R}_{\mathrm{H}} \:=\:\mathrm{1}.\mathrm{1}\:×\:\mathrm{10}^{\mathrm{7}} \:\mathrm{m}^{−\mathrm{1}} ,\:{c}\:=\:\mathrm{3}\:×\:\mathrm{10}^{\mathrm{8}} \:{ms}^{−\mathrm{1}} ,\right. \\ $$$$\left.{h}\:=\:\mathrm{6}.\mathrm{62}\:×\:\mathrm{10}^{−\mathrm{34}} \:\mathrm{Js}\right) \\ $$

Question Number 22044 Answers: 1 Comments: 0

Let A = {1, 2, 3, ....., n}, if a_i is the minimum element of the set A; (where A; denotes the subset of A containing exactly three elements) and X denotes the set of A_i ′s, then evaluate Σ_(A_i ∈X) a.

$$\mathrm{Let}\:{A}\:=\:\left\{\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:.....,\:{n}\right\},\:\mathrm{if}\:{a}_{{i}} \:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{minimum}\:\mathrm{element}\:\mathrm{of}\:\mathrm{the}\:\mathrm{set}\:{A};\:\left(\mathrm{where}\right. \\ $$$${A};\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{subset}\:\mathrm{of}\:{A}\:\mathrm{containing} \\ $$$$\left.\mathrm{exactly}\:\mathrm{three}\:\mathrm{elements}\right)\:\mathrm{and}\:{X}\:\mathrm{denotes} \\ $$$$\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:{A}_{{i}} '\mathrm{s},\:\mathrm{then}\:\mathrm{evaluate}\:\underset{{A}_{{i}} \in{X}} {\sum}{a}. \\ $$

Question Number 22043 Answers: 1 Comments: 0

In how many ways we can choose 3 squares on a chess board such that one of the squares has its two sides common to other two squares?

$$\mathrm{In}\:\mathrm{how}\:\mathrm{many}\:\mathrm{ways}\:\mathrm{we}\:\mathrm{can}\:\mathrm{choose}\:\mathrm{3} \\ $$$$\mathrm{squares}\:\mathrm{on}\:\mathrm{a}\:\mathrm{chess}\:\mathrm{board}\:\mathrm{such}\:\mathrm{that}\:\mathrm{one} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{squares}\:\mathrm{has}\:\mathrm{its}\:\mathrm{two}\:\mathrm{sides}\:\mathrm{common} \\ $$$$\mathrm{to}\:\mathrm{other}\:\mathrm{two}\:\mathrm{squares}? \\ $$

Question Number 22042 Answers: 1 Comments: 0

Determine the number of ordered pairs of positive integers (a, b) such that the least common multiple of a and b is 2^3 ∙5^7 ∙11^(13) .

$$\mathrm{Determine}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ordered} \\ $$$$\mathrm{pairs}\:\mathrm{of}\:\mathrm{positive}\:\mathrm{integers}\:\left({a},\:{b}\right)\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{least}\:\mathrm{common}\:\mathrm{multiple}\:\mathrm{of}\:{a} \\ $$$$\mathrm{and}\:{b}\:\mathrm{is}\:\mathrm{2}^{\mathrm{3}} \centerdot\mathrm{5}^{\mathrm{7}} \centerdot\mathrm{11}^{\mathrm{13}} . \\ $$

Question Number 22041 Answers: 0 Comments: 0

On the modified chess board 10 × 10, Amit and Suresh two persons which start moving towards each other. Each person moving with same constant speed. Amit can move only to the right and upwards along the lines while Suresh can move only to the left or downwards along the lines of the chess boards. The total number of ways in which Amit and Suresh can meet at same point during their trip.

$$\mathrm{On}\:\mathrm{the}\:\mathrm{modified}\:\mathrm{chess}\:\mathrm{board}\:\mathrm{10}\:×\:\mathrm{10}, \\ $$$$\mathrm{Amit}\:\mathrm{and}\:\mathrm{Suresh}\:\mathrm{two}\:\mathrm{persons}\:\mathrm{which} \\ $$$$\mathrm{start}\:\mathrm{moving}\:\mathrm{towards}\:\mathrm{each}\:\mathrm{other}.\:\mathrm{Each} \\ $$$$\mathrm{person}\:\mathrm{moving}\:\mathrm{with}\:\mathrm{same}\:\mathrm{constant} \\ $$$$\mathrm{speed}.\:\mathrm{Amit}\:\mathrm{can}\:\mathrm{move}\:\mathrm{only}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{right}\:\mathrm{and}\:\mathrm{upwards}\:\mathrm{along}\:\mathrm{the}\:\mathrm{lines} \\ $$$$\mathrm{while}\:\mathrm{Suresh}\:\mathrm{can}\:\mathrm{move}\:\mathrm{only}\:\mathrm{to}\:\mathrm{the}\:\mathrm{left} \\ $$$$\mathrm{or}\:\mathrm{downwards}\:\mathrm{along}\:\mathrm{the}\:\mathrm{lines}\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{chess}\:\mathrm{boards}.\:\mathrm{The}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of} \\ $$$$\mathrm{ways}\:\mathrm{in}\:\mathrm{which}\:\mathrm{Amit}\:\mathrm{and}\:\mathrm{Suresh}\:\mathrm{can} \\ $$$$\mathrm{meet}\:\mathrm{at}\:\mathrm{same}\:\mathrm{point}\:\mathrm{during}\:\mathrm{their}\:\mathrm{trip}. \\ $$

Question Number 22040 Answers: 0 Comments: 4

The total number of non-similar triangles which can be formed such that all the angles of the triangle are integers is

$$\mathrm{The}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{non}-\mathrm{similar} \\ $$$$\mathrm{triangles}\:\mathrm{which}\:\mathrm{can}\:\mathrm{be}\:\mathrm{formed}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{all}\:\mathrm{the}\:\mathrm{angles}\:\mathrm{of}\:\mathrm{the}\:\mathrm{triangle}\:\mathrm{are} \\ $$$$\mathrm{integers}\:\mathrm{is} \\ $$

Question Number 22038 Answers: 0 Comments: 1

The symbols +, +, ×, ×, ★, •, are placed in the squares of the adjoining figure. The number of ways of placing symbols so that no row remains empty is

$$\mathrm{The}\:\mathrm{symbols}\:+,\:+,\:×,\:×,\:\bigstar,\:\bullet,\:\mathrm{are} \\ $$$$\mathrm{placed}\:\mathrm{in}\:\mathrm{the}\:\mathrm{squares}\:\mathrm{of}\:\mathrm{the}\:\mathrm{adjoining} \\ $$$$\mathrm{figure}.\:\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{ways}\:\mathrm{of}\:\mathrm{placing} \\ $$$$\mathrm{symbols}\:\mathrm{so}\:\mathrm{that}\:\mathrm{no}\:\mathrm{row}\:\mathrm{remains}\:\mathrm{empty} \\ $$$$\mathrm{is} \\ $$

Question Number 22037 Answers: 0 Comments: 2

How many 5-digit numbers from the digits {0, 1, ....., 9} have? (i) Strictly increasing digits (ii) Strictly increasing or decreasing digits (iii) Increasing digits (iv) Increasing or decreasing digits

$$\mathrm{How}\:\mathrm{many}\:\mathrm{5}-\mathrm{digit}\:\mathrm{numbers}\:\mathrm{from}\:\mathrm{the} \\ $$$$\mathrm{digits}\:\left\{\mathrm{0},\:\mathrm{1},\:.....,\:\mathrm{9}\right\}\:\mathrm{have}? \\ $$$$\left(\mathrm{i}\right)\:\mathrm{Strictly}\:\mathrm{increasing}\:\mathrm{digits} \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{Strictly}\:\mathrm{increasing}\:\mathrm{or}\:\mathrm{decreasing} \\ $$$$\mathrm{digits} \\ $$$$\left(\mathrm{iii}\right)\:\mathrm{Increasing}\:\mathrm{digits} \\ $$$$\left(\mathrm{iv}\right)\:\mathrm{Increasing}\:\mathrm{or}\:\mathrm{decreasing}\:\mathrm{digits} \\ $$

Question Number 22036 Answers: 0 Comments: 1

2n objects of each of three kinds are given to two persons, so that each person gets 3n objects. Prove that this can be done in 3n^2 + 3n + 1 ways.

$$\mathrm{2}{n}\:\mathrm{objects}\:\mathrm{of}\:\mathrm{each}\:\mathrm{of}\:\mathrm{three}\:\mathrm{kinds}\:\mathrm{are} \\ $$$$\mathrm{given}\:\mathrm{to}\:\mathrm{two}\:\mathrm{persons},\:\mathrm{so}\:\mathrm{that}\:\mathrm{each} \\ $$$$\mathrm{person}\:\mathrm{gets}\:\mathrm{3}{n}\:\mathrm{objects}.\:\mathrm{Prove}\:\mathrm{that} \\ $$$$\mathrm{this}\:\mathrm{can}\:\mathrm{be}\:\mathrm{done}\:\mathrm{in}\:\mathrm{3}{n}^{\mathrm{2}} \:+\:\mathrm{3}{n}\:+\:\mathrm{1}\:\mathrm{ways}. \\ $$

Question Number 22035 Answers: 0 Comments: 0

The number of five digits can be made with the digits 1, 2, 3 each of which can be used atmost thrice in a number is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{five}\:\mathrm{digits}\:\mathrm{can}\:\mathrm{be}\:\mathrm{made} \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{digits}\:\mathrm{1},\:\mathrm{2},\:\mathrm{3}\:\mathrm{each}\:\mathrm{of}\:\mathrm{which}\:\mathrm{can} \\ $$$$\mathrm{be}\:\mathrm{used}\:\mathrm{atmost}\:\mathrm{thrice}\:\mathrm{in}\:\mathrm{a}\:\mathrm{number}\:\mathrm{is} \\ $$

Question Number 22023 Answers: 1 Comments: 3

Question Number 22020 Answers: 0 Comments: 0

The line of action of the resultant of two like parallel forces shifts by one fourth of the distance between the forces when the two forces are interchanged. The ratio of the two forces is

$$\mathrm{The}\:\mathrm{line}\:\mathrm{of}\:\mathrm{action}\:\mathrm{of}\:\mathrm{the}\:\mathrm{resultant}\:\mathrm{of} \\ $$$$\mathrm{two}\:\mathrm{like}\:\mathrm{parallel}\:\mathrm{forces}\:\mathrm{shifts}\:\mathrm{by}\:\mathrm{one} \\ $$$$\mathrm{fourth}\:\mathrm{of}\:\mathrm{the}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{the} \\ $$$$\mathrm{forces}\:\mathrm{when}\:\mathrm{the}\:\mathrm{two}\:\mathrm{forces}\:\mathrm{are} \\ $$$$\mathrm{interchanged}.\:\mathrm{The}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{the}\:\mathrm{two} \\ $$$$\mathrm{forces}\:\mathrm{is} \\ $$

Question Number 22014 Answers: 0 Comments: 0

The number of solution(s) of the equation ∣log_e (∣x∣)∣ + ∣x∣ = 10 is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{solution}\left(\mathrm{s}\right)\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mid\mathrm{log}_{{e}} \left(\mid{x}\mid\right)\mid\:+\:\mid{x}\mid\:=\:\mathrm{10}\:\mathrm{is} \\ $$

Question Number 22001 Answers: 1 Comments: 0

if determinant (((z−(z/4))),())=2 then value of determinant ((z),())

$${if}\:\begin{vmatrix}{{z}−\frac{{z}}{\mathrm{4}}}\\{}\end{vmatrix}=\mathrm{2}\:{then}\:{value}\:{of}\:\begin{vmatrix}{{z}}\\{}\end{vmatrix} \\ $$

Question Number 21994 Answers: 0 Comments: 1

Suppose N is an n-digit positive integer such that (a) all the n-digits are distinct; and (b) the sum of any three consecutive digits is divisible by 5. Prove that n is at most 6. Further, show that starting with any digit one can find a six-digit number with these properties.

$$\mathrm{Suppose}\:{N}\:\mathrm{is}\:\mathrm{an}\:{n}-\mathrm{digit}\:\mathrm{positive} \\ $$$$\mathrm{integer}\:\mathrm{such}\:\mathrm{that} \\ $$$$\left(\mathrm{a}\right)\:\mathrm{all}\:\mathrm{the}\:{n}-\mathrm{digits}\:\mathrm{are}\:\mathrm{distinct};\:\mathrm{and} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{any}\:\mathrm{three}\:\mathrm{consecutive} \\ $$$$\mathrm{digits}\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{5}. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:{n}\:\mathrm{is}\:\mathrm{at}\:\mathrm{most}\:\mathrm{6}.\:\mathrm{Further}, \\ $$$$\mathrm{show}\:\mathrm{that}\:\mathrm{starting}\:\mathrm{with}\:\mathrm{any}\:\mathrm{digit}\:\mathrm{one} \\ $$$$\mathrm{can}\:\mathrm{find}\:\mathrm{a}\:\mathrm{six}-\mathrm{digit}\:\mathrm{number}\:\mathrm{with}\:\mathrm{these} \\ $$$$\mathrm{properties}. \\ $$

Question Number 21990 Answers: 1 Comments: 1

i still search about a general and complete solution about this determine x in N where 7 divise 2^x +3^x note = it is just an exercise in secondary so dont go away... maybe we must use separation of cases methode....

$${i}\:{still}\:{search}\:{about}\:{a}\:{general}\:{and}\: \\ $$$${complete}\:{solution}\:{about}\:{this} \\ $$$${determine}\:{x}\:{in}\:{N}\:{where}\:\mathrm{7}\:{divise}\:\mathrm{2}^{{x}} +\mathrm{3}^{{x}} \\ $$$${note}\:=\:{it}\:{is}\:{just}\:{an}\:{exercise}\:{in}\:{secondary} \\ $$$${so}\:{dont}\:{go}\:{away}... \\ $$$${maybe}\:{we}\:{must}\:{use}\:{separation}\:{of}\:{cases} \\ $$$${methode}.... \\ $$

Question Number 21973 Answers: 1 Comments: 1

ΔABC with sides a, b, c ∈ Z and ∠A = 3∠B If its circumference is minimum, then find a, b, c

$$\Delta{ABC}\:\mathrm{with}\:\mathrm{sides}\:{a},\:{b},\:{c}\:\in\:\mathbb{Z}\:\mathrm{and}\:\angle{A}\:=\:\mathrm{3}\angle{B} \\ $$$$\mathrm{If}\:\mathrm{its}\:\mathrm{circumference}\:\mathrm{is}\:\mathrm{minimum},\:\mathrm{then} \\ $$$$\mathrm{find}\:{a},\:{b},\:{c} \\ $$

Question Number 21970 Answers: 1 Comments: 0

integrate ∫sec^3 xdx

$${integrate} \\ $$$$\int{sec}^{\mathrm{3}} {xdx} \\ $$

Question Number 21967 Answers: 1 Comments: 1

A block is tied with a thread of length l and moved in a horizontal circle on a rough table. Coefficient of friction between block and table is μ = 0.2. Find tan θ, where θ is the angle between acceleration and frictional force at the instant when speed of particle is v = (√(1.6lg))

$$\mathrm{A}\:\mathrm{block}\:\mathrm{is}\:\mathrm{tied}\:\mathrm{with}\:\mathrm{a}\:\mathrm{thread}\:\mathrm{of}\:\mathrm{length}\:{l} \\ $$$$\mathrm{and}\:\mathrm{moved}\:\mathrm{in}\:\mathrm{a}\:\mathrm{horizontal}\:\mathrm{circle}\:\mathrm{on}\:\mathrm{a} \\ $$$$\mathrm{rough}\:\mathrm{table}.\:\mathrm{Coefficient}\:\mathrm{of}\:\mathrm{friction} \\ $$$$\mathrm{between}\:\mathrm{block}\:\mathrm{and}\:\mathrm{table}\:\mathrm{is}\:\mu\:=\:\mathrm{0}.\mathrm{2}. \\ $$$$\mathrm{Find}\:\mathrm{tan}\:\theta,\:\mathrm{where}\:\theta\:\mathrm{is}\:\mathrm{the}\:\mathrm{angle} \\ $$$$\mathrm{between}\:\mathrm{acceleration}\:\mathrm{and}\:\mathrm{frictional} \\ $$$$\mathrm{force}\:\mathrm{at}\:\mathrm{the}\:\mathrm{instant}\:\mathrm{when}\:\mathrm{speed}\:\mathrm{of} \\ $$$$\mathrm{particle}\:\mathrm{is}\:{v}\:=\:\sqrt{\mathrm{1}.\mathrm{6}{lg}} \\ $$

Question Number 21962 Answers: 0 Comments: 2

Let A be a set of 16 positive integers with the property that the product of any two distinct numbers of A will not exceed 1994. Show that there are two numbers a and b in A which are not relatively prime.

$$\mathrm{Let}\:{A}\:\mathrm{be}\:\mathrm{a}\:\mathrm{set}\:\mathrm{of}\:\mathrm{16}\:\mathrm{positive}\:\mathrm{integers} \\ $$$$\mathrm{with}\:\mathrm{the}\:\mathrm{property}\:\mathrm{that}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of} \\ $$$$\mathrm{any}\:\mathrm{two}\:\mathrm{distinct}\:\mathrm{numbers}\:\mathrm{of}\:{A}\:\mathrm{will} \\ $$$$\mathrm{not}\:\mathrm{exceed}\:\mathrm{1994}.\:\mathrm{Show}\:\mathrm{that}\:\mathrm{there}\:\mathrm{are} \\ $$$$\mathrm{two}\:\mathrm{numbers}\:{a}\:\mathrm{and}\:{b}\:\mathrm{in}\:{A}\:\mathrm{which}\:\mathrm{are} \\ $$$$\mathrm{not}\:\mathrm{relatively}\:\mathrm{prime}. \\ $$

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