Remainder Problem - 2

Method-1 : \[\begin{gathered} R\left( {\frac{{{3^{98}}}}{5}} \right) = ? \hfill \\ = R\left( {\frac{{{3^{2 \times 49}}}}{5}} \right) \hfill \\ = R\left( {\frac{{{{\left( {{3^2}} \right)}^{49}}}}{5}} \right) \hfill \\ = R\left( {\frac{{{{\left( { - 1} \right)}^{49}}}}{5}} \right) \hfill \\ = R\left( {\frac{{ - 1}}{5}} \right) \hfill \\ = R\left( {\frac{{5 - 1}}{5}} \right) \hfill \\ = 4 \hfill \\ \therefore R\left( {\frac{{{3^{98}}}}{5}} \right) = 4 \hfill \\ \end{gathered} \] Method- 2: We will be using Fermat’s Little Theorem to solve this problem: \[\begin{gathered} {a^{p - 1}} \equiv \left( 1 \right)\,\bmod \,\left( p \right),p:prime \hfill \\ {3^4} \equiv \left( 1 \right)\,\bmod \,\left( 5 \right) \hfill \\ \Rightarrow {\left( {{3^4}} \right)^{24}} \equiv {\left( 1 \right)^{24}}\,\bmod \,\left( 5 \right) \hfill \\ \Rightarrow \left( {{3^{96}}} \right) \equiv \left( 1 \right)\,\bmod \,\left( 5 \right) \hfill \\ \Rightarrow \left( {{3^{96}} \times {3^2}} \right) \equiv \left( {1 \times {3^2}} \right)\,\bmod \,\left( 5 \right) \hfill \\ \Rightarrow {3^{98}} \equiv \left( 9 \right)\,\bmod \,\left( 5 \right) \hfill \\ \Rightarrow {3^{98}} \equiv \left( 4 \right)\,\bmod \,\left( 5 \right) \hfill \\ \therefore R\left( {\frac{{{3^{98}}}}{5}} \right) = 4 \hfill \\ \end{gathered} \]

Remainder Problem -1

$$\eqalign{ & R\left( {\frac{{{{18}^{201}}}}{7}} \right) = ? \cr & = R\left( {\frac{{{6^{201}} \times {3^{201}}}}{7}} \right) \cr & = R\left( {\frac{{{{\left( { - 1} \right)}^{201}} \times {{\left( {{3^3}} \right)}^{67}}}}{7}} \right) \cr & = R\left( {\frac{{ - 1 \times {{\left( { - 1} \right)}^{67}}}}{7}} \right) \cr & = R\left( {\frac{{ - 1 \times - 1}}{7}} \right) \cr & = R\left( {\frac{1}{7}} \right) \cr & = 1 \cr} $$

Divisibility By Eleven

Divisibility By Eleven

Number Theory- Factorials

Tech Mahindra Placement Questions

   Here are the important  Tech Mahindra Placement Questions.

Section: Quantitative Aptitude – Arithmetic Ability

Choose the correct option.

Q.1 ) A man is three (3) years older than his wife and four (4) times as old as his son. If the son attains an age of fifteen (15) years after three (3) years, what is the present age of the mother?

A: 60 years          B: 51 years          C: 48 years          D: 45 years          E: 65 years

Q.2 ) If Thursday was the day after the day before yesterday five days ago, what is the least number of days ago when sunday was three days before the day after tomorrow?

A: Two days ago               B: Three days ago            C: Four days ago               D: Five days ago                E: None of these

Q.3 ) A train started from station ‘A’ and proceeded towards station ‘B’ at a speed of 48 Km/hr. Forty-five minutes later another train started from station ‘B’ and proceeded towards station ‘A’ at 50 Km/hr. If the distance between the two stations is 232 Km, at what distance from station ‘A’ will the trains meet?

A: 132 Km            B: 144 Km            C: 108 Km            D: 160 Km            E: None of these

Q.4)  If a number is five time as great as another number which is four less than forty, then the number is:

A: 220                    B: 180                    C: 144                    D: 200                    E: 196

Q.5 ) How many such 3’s are there in the following number sequence which are immediately Preceded by an odd number and immediately followed by an even number? 5 3 8 9 4 3 7 2 3 8 1 3 8 4 2 3 5 7 3 4 2 3 6

A: One                  B: Two                  C: Three               D: Four                                 E: More than four

Q.6 ) The sum of two consecutive number is 87. Which is the larger number?

A: 42                      B: 43                      C: 44                      D: 45                      E: 46

Q.7 ) If ratio of profit of A and B is 4:5 and they together invested Rs.90,000 then money invested by B is

A: Rs.48,000        B: Rs.45,000        C: Rs.30,000        D: Rs.50,000       

Q.8 ) If 10 litres of an oil of Rs.50 per litres be mixed with 5 litres of another oil of Rs.66 per litre then what is the rate of mixed oil per litre?

A: Rs.49.17          B: Rs.51.03          C: Rs.54.17          D: Rs.55.33         

Q.9 ) A runs twice as fast as B and gives B a start of 50m.How long should the racecourse be so that A and B might reach in the same time?

A: 75 m.                                B: 80 m.                C: 150 m.              D: 100 m.            

Q.10) Sum of ages of father and his son is 60 years while difference is 30 years. what is father’s age?

A: 60 years                          B: 45 years          C: 50 years          D: 40 years         

Q.11) The average of all odd numbers upto 100 is :

A: 49                                      B: 49.5                   C: 50                      D: 51     

Q.12 ) The largest 5-digit number exactly divisible by 91 is

A: 99921               B: 99918               C: 99981               D: 99971               E: None of these

Q.13 ) My age becomes half that of my brother`s if we simply add 2 years to his present age. If I am 25 years old today, my brother will be

A: 46      B: 48      C: 44      D: 36      E: 38

Q.14) The ratio between the present ages of A and B is 5 : 3 respectively. The ratio between A`s age 4 years ago and B`s age 4 years hence is 1 : 1. What is the ratio between A`s age 4 years hence and B`s age 4 years ago ?

A: 1:3     B: 2:1     C: 3:1     D: 4:1     E: None of these

Q.15)  Find the sub-triplicate ratio of [8x]^3 :64y^3

A: x : 2y                B: x^2 : 4y^2       C: 4y^2 : x^2       D: 2y : x               

  Section: Verbal Reasoning – General

In each of the following series determine the order of the letters. Then from the given options select the one which will complete the given series.

Q.1)  DUE ,CVV,BWU,AXT,?

A: BYS   B: CZV   C: XYS    D: ZYS  

Which letter(s) in each of the following series is wrong or is misfit in the series ?

Q.2)  Z T P K H F

A: Z        B: P        C: T         D: F       

Instructions for Question No. 3 :

In the following questions select the right option which indicates the correct code for the word or letter given in the question.

Q.3) If PHILOSOPHY is coded as HPLISOPOYH, ORNAMENTAL will be coded as :

A: ROANEMNTLA             B: ONRAMNEALT             C: ROANEMTNLA             D: ROANEMNATL            

Instructions for Question No. 4 to 5 :

In the following questions study the coded patterns and then select the right option from the given alternatives.

Q.4) In a certain code language,(1)’lo ni hie pun’stands for’he is drinking coke’;(2)’hol ful gui pun’stands for’she is eating food’;and(3)’ne ful ni lo’stands for ‘drinking coke and food’.Which of the following words is the code for ‘he’?

A: hie                    B: lo                       C: pun                   D: ni      

Q.5) In a certain code language,’col tip mot’means’singing is appreciable’,’mot baj min’means ‘dancing is good’and’tip nop baj’means ‘singing and dancing’.Which of the following means ‘Good’in that code language?

A: mot                  B: min                   C: baj                     D: cannot be determined            

Instructions for Question No. 6 to 7 :

In each of the following questions keenly study the relationship mentioned between the persons, and then from the given options select the right relationship as the answer.

Q.6) Pointing to Suman,Amit said,”He is my sister’s only brother’s son”.How is Suman related to Amit?

A: Grandson       B: Son   C: Nephew         D: Cannot be determined            E: None of these

Q.7 ) B is maternal uncle of A. C is maternal grandfather of B. D is grandson of C. How is D related to A?

A: Maternal uncle           B: Cousin brother             C: Nephew         D: Maternal grandfather               E: None of these

Instructions for Question No. 8 to 9 :

Choose the correct option.

Q.8 ) In a telephone directory,which of the following names will appear in the middle?

A: Randhir           B: Randesh         C: Rama                D: Ramesh          E: Renmurthi

Q.9 ) Which one would be a meaningful order of the following?

1. Snake 2. Grass 3. Eagle 4. Frog 5. Insect

A: 3 2 1 4 5           B: 5 2 1 4 3           C: 2 5 4 1 3           D: 2 4 5 3 1           E: 2 4 5 1 3

Instructions for Question No. 10 :

Find out the one word among the options which cannot be formed by using the letters of the word as given in each question.

Q.10)  COMMENTATOR

A: COMMON     B: MOMENT       C: COSMOS         D: TART

Instructions for Question No. 11 to 12 :

In the following questions select the number(s) from the given options for completing the given series.

Q.11)  3,5,9,15,23,?,45

A: 37      B: 35      C: 31      D: 33     

Q.12)   0,2,10,30,68,?

A: 125    B: 120    C: 175    D: 130   

Instructions for Question No. 13 :

In the following questions, select the number from the given options which follows the same relationship as shared between the first two numbers.

Q.) 13       92 : 69 :: 46

A: 57      B: 23      C: 31      D: 19     

Instructions for Question No. 14 to 15 :

In the questions given below establish the relationship between two words. Then from the given option select the one which has the same relationship as of the given two words.

Q.14 )  SYMPATHY : EQUITY

A: Common : Exclusive B: Certain : Precise          C: Snub : Admire              D: Beloved : Paltry           E: Unity : Strife

Q.15)  JUPITER : PLANET

A: Earth : Orbit B: Saturn : Sun   C: Cosmos : Star                D: Moon : Satellite          E: Milky way : Pole

Instructions for Question No. 16 to 20 :

In each of the following questions four words are alike in some manner. Spot the odd one out.

Q.16)       

A: Van   B: Aeroplane      C: Helicopter      D: Transport       E: Scooter

Q.17)       

A: Horse               B: Cow C: Donkey           D: Pig     E: Turkey

Q.18)       

A: Picture            B: Poster              C: Photograph   D: Scenery          E: Painting

Q.19)       

A: Kite   B: Vulture            C: Hawk                D: Bat    E: Duck

Q.20)      

A: Tool B: Rule C: Cool D: Pool E: Fool

Instructions for Question No. 21 :

Directions : Choose the correct answer :

Q.21)  BLOOD is related to ARTERY in the same way as WATER is related to

A: Vein                                 B: Pipe                  C: Aorta                D: Straw              

Instructions for Question No. 22 :

Directions : Find out the wrong number in the series

Q.22) 1,2,8,33,148,760,4626

A: 2        B: 8         C: 33      D: 148    E: 760

Instructions for Question No. 23 :

Directions : Insert the missing number

Q.23) 10,5,13,10,16,20,19,(……)

A: 22      B: 40      C: 38      D: 23     

Instructions for Question No. 24 :

Directions : In each of the following questions, there is some relationship between the two terms to the left of : : and the same relationship holds between the two terms to its right. Also, in each question, one term either to the right of : : of to the left of it is missing. This term is given as one of the alternatives given below each question. Find out this term.

Q.24 ) KORT is related to PJWO in the same way as FINR is related to …….?……….

A: KCSM               B: KDSM               C: JSMR                D: JCRN               

Instructions for Question No. 25 :

Directions: In each questions below, two pairs of numbers are given but one number in the second pair is missing. Identify the relationship between the two numbers in the first pair and find the missing number in the second pair such that the numbers in second pair also follow the same relationship.

Q.25) 25: 125 :: 64 : ?

A: 521    B: 512    C: 513    D: 520   

Instructions for Question No. 26:

Directions: Each of the following questions has a pair of CAPITALIZED words followed by four pairs of words. Choose the pair of words which best expresses the relationship similar to that expressed in the capitalized pair.

Q.26) SPEAR : DART

A: knife : sword                B: door : window              C: mountain : molehill    D: cannon : gun                

Instructions for Question No. 27 :

Directions : In each of these questions, only one of the four alternatives given under each question satisfies the same relationship as is found between two terms on the left side of the sign (::). Find the correct alternative to fill in the right side of the sign (::).

Q.27) Spasm : Pain :: ____________________ : ____________________

A: Touch : Feel B: Fast : Quick    C: Flash : Light   D: Water : Wet

Instructions for Question No. 28 to 29 :

Directions : In each of the following questions, five words have been given out of which four are alike in some manner, while the fifth one is different. Choose out the odd one.

Q.28) Choose the odd one out

A: Flute                B: Guitar              C: Sitar D: Violin               E: Veena

Q.29) Choose the odd one out

A: Sahara             B: Thar C: Gobi                 D: Sunderban    E: Kalahari

Instructions for Question No. 30 to 31 :

Directions : In each of the following questions, four words have been given, out of which three are alike in some manner and the fourth one is different. Choose out the odd one.

Q.30) Choose the odd one out

A: Platform         B: Dock                 C: Bus-stand       D: Park

Instructions for Question No. 31 to 30 :

Directions : Choose the odd numeral pair/group in each of the following questions:

Q.31 )      

A: 48:134              B: 40:110              C: 18:48                D: 30:80               

Instructions for Question No. 32 to 31 :

Directions : In each of the following questions, certain pairs of words are given, out of which the words in all pairs except one, bear a certain common relationship. Choose the pair in which the words are differently related.

Q.32      

A: Mercury : Sun              B: Moon : Earth                 C: Star : Galaxy D: Wheel : Axle                

Instructions for Question No. 33 to 36 :

Directions: In each of the following questions, five words have been given out of which four are alike in some manner, while the fifth one is different. Choose out the odd one.

Q.33)  Choose the odd one out

A: Feathers         B: Tentacles        C: Scales              D: Pseudopodia                E: Flagella

Q.34) Choose the odd one out

A: Kiwi B: Eagle                C: Emu D: Penguin          E: Ostrich

Q.35) Choose the odd one out

A: Mustard         B: Rapeseed       C: Sesame           D: Cashewnut    E: Groundnut

Q.36) Choose the odd one out

A: Indigo              B: Orange            C: Yellow              D: Pink E: Green

Instructions for Question No. 37 to 38 :

Directions : In each of the following questions, four words have been given, out of which three are alike in some manner and the fourth one is different. Choose out the odd one.

Q.37)  Choose the odd one out

A: Kleptomania                B: Schizophrenia               C: Agoraphobia                 D: Alzheimer’s disease

Instructions for Question No. 38 :

Directions : Read the information carefully and answer the questions:

M,N,O,P,Q,R,T and V are eight sports persons. Three of them play cricket, two of them play hockey, one play badminton and two play volleyball. There are three lady members among them. No lady either plays hockey or badminton and at least one lady plays the remaining two games. None of the games is played by only ladies. P plays badminton Q`s sister V plays cricket. M plays volleyball with T. N plays cricket. O and T are sisters.

Q.38 ) Which of the following persons play cricket ?

A: NRV B: NQV                 C: MPV                 D: Data inadequate         E: None of these

Instructions for Question No. 39 :

Directions : Read the following information to answer the given questions:

A bag contains coins of four different denominations, viz. 1 rupee, 50-paise, 25-paise and 10-paise. There are as many 50-paise coins as the value of 25-paise coins in rupees. The value of 1-rupee coins is 5 times the value of 50-paise coins. The ratio of the number of 10-paise coins to that of 1-rupee coins is 4 : 3, while the total number of coins in the bag is 325.

Q.39       How many 10-paise coins are there?

A: 25      B: 50      C: 75      D: 100   

Instructions for Question No. 40 :

Directions : study the following information carefully and answer the question.

B, M, T, R, K, H and D are travelling in a train compartment with III-tier sleeper berth. Each of them is of a different profession of engineer,doctor,architect,pharmacist,lawyer,journalist and pathologist. They occupied two lower berths,three middle berths and two upper berths. B,the engineer is not on the upper berth. The architect is the only other person who occupies the same type of berth so that B,M and H are not on the middle berth and their professions are pathologist and lawyer respectively. T is a pharmacist. D is neither a journalist nor an architect. K occupies same type of berths as that of the doctor.

Q.40) What is D`s profession ?

A: Doctor             B: Engineer         C: Lawyer            D: Pharmacist    E: Data inadequate

 Section: Verbal Reasoning – Paragraph

Paragraph : 1

(i) A,B,C,D,E,F and G are sitting on a wall and all of them are facing east. (ii) C is on the immediate right of D. (iii) B is at an extreme end and has E as his neighbour. (iv) G is between E and F. (v) D is sitting third from the south end.

Please refer paragraph 1 for the next question

Q.1         Which of the following pairs of people are sitting at the extreme ends?

A: AE     B: AB     C: FB      D: CB     E: Cannot be determined

Please refer paragraph 1 for the next question

Q.2         Immediately between which of the following pairs of people is D sitting?

A: CE      B: AC     C: CF      D: AF     E: None of these

Please refer paragraph 1 for the next question

Q.3         Name the person who should change places with C such that he gets the third place from the north end?

A: G       B: F         C: E         D: Can’t be determined                 E: None of these

Please refer paragraph 1 for the next question

Q.4         Who is sitting to the right of E?

A: F        B: D        C: C        D: A        E: None of these

Please refer paragraph 1 for the next question

Q.5         Which of the following pairs of people are sitting next to F?

A: EG     B: DC     C: ED      D: GD    E: None of these

               

 

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