__Numbers__**1) 2^2n-1 is always divisible by 3**

2^2n-1 = (3-1)^2n -1

= 3M +1 -1

= 3M, thus divisible by 3

**2) What is the sum of the divisors of 2^5.3^7.5^3.7^2?**

ANS : (2^6-1)(3^8-1)(5^4-1)(7^3-1)/2.4.6

Funda : if a number 'n' is represented as

a^x * b^y * c^z ....

where, {a,b,c,.. } are prime numbers then

Quote: |

(a) the total
number of factors is (x+1)(y+1)(z+1) .... (b) the total number of relatively prime numbers less than the number is n * (1-1/a) * (1-1/b) * (1-1/c).... (c) the sum of relatively prime numbers less than the number is n/2 * n * (1-1/a) * (1-1/b) * (1-1/c).... (d) the sum of factors of the number is {a^(x+1)} * {b^(y+1)} * ...../(x*y*...) |

**3) what is the highest power of 10 in 203!**ANS : express 10 as product of primes; 10 = 2*5

divide 203 with 2 and 5 individually

203/2 = 101

101/2 = 50

50/2 = 25

25/2 = 12

12/2 = 6

6/2 = 3

3/2 = 1

thus power of 2 in 203! is, 101 + 50 + 25 + 12 + 6 + 3 + 1 = 198

divide 203 with 5

203/5 = 40

40/5 = 8

8/5 = 1

thus power of 5 in 203! is, 49

so the power of 10 in 203! factorial is 49

**4) x + y + z = 7 and xy + yz + zx = 10, then what is the maximum value of x? ( CAT 2002 has similar question )**

ANS: 49-20 = 29, now if one of the y,z is zero, then the sum of other 2 squares shud be equal to 29, which means, x can take a max value at 5

**5) In how many ways can 2310 be expressed as a product of 3 factors?**

ANS: 2310 = 2*3*5*7*11

When a number can be expressed as a product of n distinct primes,

then it can be

**expressed as a product of 3 numbers in (3^(n+1) + 1)/2 ways**

**6) In how many ways, 729 can be expressed as a difference of 2 squares?**

ANS: 729 = a^2 - b^2

= (a-b)(a+b),

since 729 = 3^5,

total ways of getting 729 are, 1*729, 3*243, 9*81, 27*27.

So 4 ways

Funda is that, all four ways of expressing can be used to findout distinct a,b values,

for example take 9*81

now since 9*81 = (a-b)(a+b) by solving the system a-b = 9 and a+b = 81 we can have 45,36 as soln.

**7) How many times the digit 0 will appear from 1 to 10000**

ANS: In 2 digit numbers : 9,

In 3 digit numbers : 18 + 162 = 180,

In 4 digit numbers : 2187 + 486 + 27 = 2700,

total = 9 + 180 + 2700 + 4 = 2893

**8 ) What is the sum of all irreducible factors between 10 and 20 with denominator as 3?**

ANS :

sum = 10.33 + 10.66 + 11.33 + 11.66 + 12.33 + 12.66 + 13.33 + 13.66…….

= 21 + 23 + ……

= 300

**9) if n = 1+x where x is the product of 4 consecutive number then n is,**

1) an odd number,

2) is a perfect square

1) an odd number,

2) is a perfect square

SOLN : (1) is clearly evident

(2) let the 4 numbers be n-2,n-1,n and n+1 then by multing the whole thing and adding 1 we will have a perfect square

**10) When 987 and 643 are divided by same number 'n' the reminder is also same, what is that number if the number is a odd prime number?**

ANS : since both leave the same reminder, let the reminder be 'r',

then, 987 = an + r

and 643 = bn + r and thus

987 - 643 is divisible by 'r' and

987 - 643 = 344 = 86 * 4 = 43 * 8 and thus the prime is 43

hence 'r' is 43

**11) when a number is divided by 11,7,4 the reminders are 5,6,3 respectively. what would be the reminders when the same number is divided by 4,7,11 respectively?**

ANS : whenever such problem is given,

we need to write the numbers in top row and rems in the bottom row like this

11 7 4

| \ \

5 6 3

( coudnt express here properly )

now the number is of the form, LCM ( 11,7,4 ) + 11*(3*7 + 6) + 5

that is 302 + LCM(11,7,4) and thus the rems when the same number is divided by 4,7,11 respectively are,

302 mod 4 = 2

75 mod 7 = 5

10 mod 11 = 10

**12) a^n - b^n is always divisible by a-b**

**13) if a+b+c = 0 then a^3 + b^3 + c^3 = 3abc**

EXAMPLE: 40^3-17^3-23^3 is divisble by

since 40-23-17 = 0, 40^3-17^3-23^3 = 3*40*23*17 and thus, the number is divisible by 3,5,8,17,23 etc.

**14) There is a seller of cigerette and match boxes who sits in the narrow lanes of cochin. He prices the cigerattes at 85 p, but found that there are no takers. So he reduced the price of cigarette and managed to sell all the cigerattes, realising Rs. 77.28 in all. What is the number of cigerattes?**

a) 49

b) 81

c) 84

d) 92

ANS : (d)

since 77.28 = 92 * 84, and since price of cigarette is less than 85, we have (d) as answer

Quote: |

i have given this question to make the funda clear |

**15) What does 100 stand for if 5 X 6 = 33**

ANS : 81

SOLN : this is a number system question,

30 in decimal system is 33 in some base 'n', by solving we will have n as 9

and thus, 100 will be 9^2 = 81

**16) In any number system 121 is a perfect square,**

SOLN: let the base be 'n'

then 121 can be written as n^2 + 2*n + 1 = (n+1)^2

hence proved

**17) Most of you ppl know these, anyways, just in case**

Quote: |

(a) sum of
first 'n' natural numbers - n*(n+1)/2 (b) sum of the squares of first 'n' natural numbers - n*(n+1)*(2n+1)/6 (c) sum of the cubes of first 'n' natural numbers - n^2*(n+1)^2/4 (d) total number of primes between 1 and 100 - 25 |

**18 ) See Attachment**

**to know how to find LCM, GCF of Fractions**

Quote: |

CAT 2002 has
2 questions on the above simple concept |

**19) Converting Recurring Decimals to Fractions**

let the number x be 0.23434343434........

thus 1000 x = 234.3434343434......

and 10 x = 2.3434343434.........

thus, 990 x = 232

and hence, x = 232/990

20)

**Reminder Funda**

**(a) (a + b + c) % n = (a%n + b%n + c%n) %n**

EXAMPLE: The reminders when 3 numbers 1221, 1331, 1441 are divided by certain number 9 are 6, 8, 1 respectively. What would be the reminder when you divide 3993 with

9? ( never seen such question though )

the reminder would be (6 + 8 + 1) % 9 = 6

**(b) (a*b*c) % n = (a%n * b%n * c%n) %n**

EXAMPLE: What is the remainder left when 1073 * 1079 * 1087 is divided by 119 ? ( seen this kinda questions alot )

1073 % 119 = ?

since 1190 is divisible by 119, 1073 mod 119 is 2

and thus, "the remainder left when 1073 * 1079 * 1087 is divided by 119 " is 2*8*16 mod 119 and that is 256 mod 119 and that is (238 + 18 ) mod 119 and that is 18

Glossary : % stands for reminder operation

find
the number of zeroes in 1^1* 2^2* 3^3* 4^4.............. 98^98* 99^99* 100^100

the
expresion can be rewritten as (100!)^100 / 0!* 1!* 2!* 3!....99!

Now the numerator has 2400 zeros

the formular for finding number of zeros in n! is

[n/5]+[n/5^2]...[n/5^r]

where r is such that 5^r<=n<5^(r+1)

and [..] is the grestest integer function

for the numerator find the number of zeros using the above formulae..

for 0!...4! number of zeros ..0

5!...9!.number os zeros ..1

9!...14!... 2

15!..19!..................3

20!..24!..................4!

now at 25! the series makes a jump to 6

25!...29!.................6

30!...34!.................7

this goes on and again makes a jump at 50!

and then at 75!

so the number of zeros is...

5(1+2....19) + 25+ 50+ 75

the last 3 terms 25 50 and 75 are because of the jumps..

this gives numerator has 1100 zeros

now total number of zeros in expression is no of zeros in denominator - no of zeros in numerator

2400 - 1100

the Answer 1300

Now the numerator has 2400 zeros

the formular for finding number of zeros in n! is

[n/5]+[n/5^2]...[n/5^r]

where r is such that 5^r<=n<5^(r+1)

and [..] is the grestest integer function

for the numerator find the number of zeros using the above formulae..

for 0!...4! number of zeros ..0

5!...9!.number os zeros ..1

9!...14!... 2

15!..19!..................3

20!..24!..................4!

now at 25! the series makes a jump to 6

25!...29!.................6

30!...34!.................7

this goes on and again makes a jump at 50!

and then at 75!

so the number of zeros is...

5(1+2....19) + 25+ 50+ 75

the last 3 terms 25 50 and 75 are because of the jumps..

this gives numerator has 1100 zeros

now total number of zeros in expression is no of zeros in denominator - no of zeros in numerator

2400 - 1100

the Answer 1300

__SOME MORE SHORTCUTS:__

**To find out if a number is divisible by seven:**`Take the last digit, double it, and subtract it from the rest of the`

`number.`

`If the answer is more than a 2 digit number perform the above`

`again.`

`If the result is 0 or is divisible by 7 the original number is also`

`divisible by 7.`

`Example 1 ) 259`

`9*2= 18.`

`25-18 = 7 which is divisible by 7 so 259 is also divisible by 7.`

`Example 2 ) 2793`

`3*2= 6`

`279-6= 273`

`now 3*2=6`

`27-6= 21 which is divisible by 7 so 2793 is also divisible by 7 .`

`Now find out if following are divisible by 7`

`1) 2841`

`2) 3873`

`3) 1393`

`4) 2877`

**TO**FIND SQUARE OF
A NUMBER BETWEEN 40 to 50`Sq (44) .`

`1) Subtract the number from 50 getting result A.`

`2) Square A getting result X.`

`3) Subtract A from 25 getting result Y`

`4) Answer is xy`

`EXAMPLE 1 : 44`

`50-44=6`

`Sq of 6 =36`

`25-6 = 19`

`So answer 1936`

`EXAMPLE 2 : 47`

`50-47=3`

`Sq 0f 3 = 09`

`25-3= 22`

`So answer = 2209`

`NOW TRY To Find Sq of 48 ,26 and 49`

**TO**FIND SQUARE OF A 3
DIGIT NUMBER :`LET THE NUMBER BE XYZ`

`SQ (XYZ) is calculated like this`

`STEP 1. Last digit = last digit of SQ(Z)`

`STEP 2. Second Last Digit = 2*Y*Z + any carryover from STEP 1.`

`STEP 3. Third Last Digit 2*X*Z+ Sq(Y) + any carryover from STEP`

`2.`

`STEP 4. Fourth last digit is 2*X*Y + any carryover from STEP 3.`

`STEP 5 . In the beginning of result will be Sq(X) + any carryover`

`from Step 4.`

`EXAMPLE :`

`SQ (431)`

`STEP 1. Last digit = last digit of SQ(1) =1`

`STEP 2. Second Last Digit = 2*3*1 + any carryover from STEP`

`1.= 6`

`STEP 3. Third Last Digit 2*4*1+ Sq(3) + any carryover from STEP`

`2.= 2*4*1 +9= 17. so 7 and 1 carryover`

`STEP 4. Fourth last digit is 2*4*3 + any carryover (which is 1) . =`

`24+1=25. So 5 and carry over 2.`

`STEP 5 . In the beginning of result will be Sq(4) + any carryover`

`from Step 4. So 16+2 =18.`

`So the result will be 185761.`

`If the option provided to you are such that the last two digits are`

`different, then you need to carry out first two steps only , thus`

`saving time. You may save up to 30 seconds on each`

`calculations and if there are 4 such questions you save 2`

`minutes which may really affect UR Percentile score.`

**PYTHAGORAS THEROEM :**

`In any given exam there are about 2 to 3 questions based on pythagoras theorem. Wouldn't it be nice that you remember some of the pythagoras triplets thus saving up to 30 seconds in each question. This saved time may be used to attempt other questions. Remember one more right question may make a lot of difference in UR PERCENTILE score.`

`The unique set of pythagoras triplets with the Hypotenuse less than 100 or one of the side less than 20 are as follows :`

`(3,4,5), (5, 12, 13), (8, 15, 17), (7, 24, 25), (20, 21, 29), (12, 35, 37), (9, 40, 41), (28, 45, 53), (11, 60, 61), (33, 56, 65), (16, 63, 65), (48, 55, 73), (36, 77, 85), (13, 84, 85), (39, 80, 89), and (65, 72, 97).`

`(15,112,113), (17,144,145), (19,180,181), (20,99,101)`

`If you multiply the digits of the above mentioned sets by any constant you will again get a pythagoras triplet .`

`Example : Take the set (3,4,5).`

`Multiply it by 2 you get (6,8,10) which is also a pythagoras triplet.`

`Multiply it by 3 you get ( 9,12,15) which is also a pythagoras triplet.`

`Multiply it by 4 you get (12,16,20) which is also a pythagoras triplet.`

`You may multiply by any constant you will get a pythagoras triplet`

`Take another example (5,12,13)`

`Multiply it by 5,6 and 7 and check if you get a pythagoras triplet.`

**TIPS FOR SMART GUESSING :**`You will notice that in any case, whether it is a unique triplet or it is a derived triplet (derived by multiplying a constant to a unique triplet), all the three numbers cannot be odd.`

`In case of unique triplet , the hypotenuse is always odd and one of the remaining side is odd the other one is even.`

`Below are the first few unique triplets with first number as Odd.`

`3 4 5`

`5 12 13`

`7 24 25`

`9 40 41`

`11 60 61`

`You will notice following trend for unique triplets with first side as odd.`

`Hypotenuse = (Sq(first side) +1) / 2`

`Other side = Hypotenuse -1`

`Example : First side = 3 ,`

`so hypotenuse = (3*3+1)/2= 5 and other side = 5-1=4`

`Example 2: First side = 11`

`so hypotenuse = (9*9+1)/2= 41 and other side = 41-1=40`

`Please note that the above is not true for a derived triplet for example 9,12 and 15, which has been obtained from multiplying 3 to the triplet of 3,4,5. You may check for other derived triplets.`

`Below are the first few unique triplets with first number as Even .`

`4 3 5`

`8 15 17`

`12 35 37`

`16 63 65`

`20 99 101`

`You will notice following trend for unique triplets with first side as Even.`

`Hypotenuse = Sq( first side/ 2)+1`

`Other side = Hypotenuse-2`

`Example 1. First side =8`

`So hypotenuse = sq(8/2) +1= 17`

`Other side = 17-2=15`

`Example 2. First side = 16`

`So hypotenuse = Sq(16/2) +1 =65`

`Other side = 65-2= 63`

**PROFIT AND LOSS**`: In every exam there are from one to three`

`questions on profit and loss, stating that the cost was first`

`increased by certain % and then decreased by certain %. How`

`nice it would be if there was an easy way to calculate the final`

`change in % of the cost with just one formula. It would really help`

`you in saving time and improving UR Percentile. Here is the`

`formula for the same :`

`Suppose the price is first increase by X% and then decreased`

`by Y% , the final change % in the price is given by the following`

`formula`

`Final Difference % = X- Y - XY/100.`

`EXAMPLE 1. : The price of T.V set is increased by 40 % of the`

`cost price and then decreased by 25% of the new price . On`

`selling, the profit for the dealer was Rs.1,000 . At what price was`

`the T.V sold.`

`From the above mentioned formula you get :`

`Final difference % = 40-25-(40*25/100)= 5 %.`

`So if 5 % = 1,000`

`then 100 % = 20,000.`

`C.P = 20,000`

`S.P = 20,000+ 1000= 21,000.`

`EXAMPLE 2 : The price of T.V set is increased by 25 % of cost`

`price and then decreased by 40% of the new price . On selling,`

`the loss for the dealer was Rs.5,000 . At what price was the T.V`

`sold.`

`From the above mentioned formula you get :`

`Final difference % = 25-40-(25*45/100)= -25 %.`

`So if 25 % = 5,000`

`then 100 % = 20,000.`

`C.P = 20,000`

`S.P = 20,000 - 5,000= 15,000.`

`Now find out the difference in % of a product which was :`

`First increased by 20 % and then decreased by 10 %.`

`First Increased by 25 % and then decrease by 20 %.`

`First Increased by 20 % and then decrease by 25 %.`

`First Increased by 10 % and then decrease by 10 %.`

`First Increased by 20 % and then decrease by 15 %.`

**TIPS TO IMPROVE**UR PERCENTILE :`HOW ABOUT SOLVING THE FOLLOWING QUESTION IN JUST`

`10 SECONDS`

`Ajay can finish work in 21 days and Blake in 42 days. If Ajay,`

`Blake and Chandana work together they finish the work in 12`

`days. In how many days Blake and Chandana can finish the`

`work together ?`

`(21*12 )/(24-12) = (21*12)/9= 7*4= 28 days.`

`NOW CAREFULLY READ THE FOLLOWING TO SOLVE THE`

`TIME AND WORK PROBLEMS IN FEW SECONDS.`

**TIME AND WORK :**`1. If A can finish work in X time and B can finish work in Y time`

`then both together can finish work in (X*Y)/ (X+Y) time.`

`2. If A can finish work in X time and A and B together can finish`

`work in S time then B can finish work in (XS)/(X-S) time.`

`3. If A can finish work in X time and B in Y time and C in Z time`

`then they all working together will finish the work in`

`(XYZ)/ (XY +YZ +XZ) time`

`4. If A can finish work in X time and B in Y time and A,B and C`

`together in S time then :`

`C can finish work alone in (XYS)/ (XY-SX-SY)`

`B+C can finish in (SX)/(X-S)`

`and A+ C can finish in (SY)/(Y-S)`

`Here is another shortcut to improve URPERCENTILE.`

`TYPE 1 : Price of a commodity is increased by 60 %. By how`

`much % should the consumption be reduced so that the`

`expense remain the same.`

`TYPE 2 : Price of a commodity is decreased by 60 %. By how`

`much % can the consumption be increased so that the expense`

`remain the same.`

`Solution :`

`TYPE1 : (100* 60 ) / (100+60) = 37.5 %`

`TYPE 2 : (100* 60 ) / (100-60) = 150 %`