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takes you to places where you belong basic concepts definitions and identities number system test of divisibility 1 a number is divisible by `2 if it ends in zero or in a digit which is a multiple of `2 i.e 2,4 6 8 2 a number is divisible by `3 if the sum of the digits is divisible by `3 3 a number is divisible by `4 if the number formed by the last two digits i.e tens and units are divisible by 4 4 a number is divisible by `5 if it ends in zero or 5 5 a number is divisible by `6 if it divisible by `2 as well as by `3 6 a number is divisible by `8 if the number formed by the last three digits i.e hundreds tens and units is divisible by `8 7 a number is divisible by `9 if the sum of its digit is divisible by `9 8 a number is divisible by `10 if it ends in zero 9 a number is divisible by `11 if the difference between the sums of the digits in the even and odd places is zero or a multiple of `11 lcm lcm of a given set of numbers is the least number which is exactly divisible by every number of the given set hcf hcf of a given set of numbers is the highest number which divides exactly every number of the given set lcm hcf 1 product of two numbers hcf × lcm hcf of numerators 2 hcf of fractions lcm of denominators lcm of numerators 3 lcm of fractions hcf of denominators lcm × hcf 4 one number 2nd number product of the numbers 5 lcm of two numbers hcf product of the numbers 6 hcf lcm © entranceguru.com private limited basic concepts.doc -1-

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p. 2

takes you to places where you belong examples to follow 1 the square of an odd number is always odd 2 a number is said to be a prime number if it is divisible only by itself and unity ex 1 2 3 5,7,11,13 etc 3 the sum of two odd number is always even 4 the difference of two odd numbers is always even 5 the sum or difference of two even numbers is always even 6 the product of two odd numbers is always odd 7 the product of two even numbers is always even problems 1 if a number when divided by 296 gives a remainder 75 find the remainder when 37 divides the same number method let the number be `x say x 296k 75 where `k is quotient when `x is divided by `296 37 × 8k 37 × 2 1 378k 2 1 hence the remainder is `1 when the number `x is divided by 37 2 if 232+1 is divisible by 641 find another number which is also divisible by `641 method consider 296+1 2323 13 232 +1 264-232 +1 from the above equation we find that 296+1 is also exactly divisible by 641 since it is already given that 232+1 is exactly divisible by `641 3 if m and n are two whole numbers and if mn 25 find nm given that n 1 method © entranceguru.com private limited basic concepts.doc -2-

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p. 3

takes you to places where you belong mn 25 52 m 5 n 2 nm 25 32 4 find the number of prime factors of 610 × 717 × 5527 610 × 717 × 5527 210×310×717×527×1127 the number of prime factors the sum of all the indices viz 10 10 17 27 27 91 5 a number when successively divided by 9 11 and 13 leaves remainders 8 9 and 8 respectively method the least number that satisfies the condition 8 9×9 8×9×11 8 81 792 881 6 a number when divided by 19 gives the quotient 19 and remainder 9 find the number let the number be `x say x 19 × 19 9 361 9 370 7 four prime numbers are given in ascending order of their magnitudes the product of the first three is 385 and that of the last three is 1001 largest of the given prime numbers the product of the first three prime numbers 385 the product of the last three prime numbers 1001 in the above products the second and the third prime numbers occur in common the product of the second and third prime numbers hcf of the given products hcf of 385 and 1001 77 largest of the given primes 1001 13 77 find the © entranceguru.com private limited basic concepts.doc -3-

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p. 4

takes you to places where you belong square root cube root surds and indices characteristics of square numbers 1 a square cannot end with an odd number of zeros 2 a square cannot end with an odd number 2 3 7 or 8 3 the square of an odd number is odd 4 the square of an even number is even 5 every square number is a multiple of 3 or exceeds a multiple of 3 by unity ex 4 × 4 16 5 × 3 1 5 × 5 25 8 × 3 1 7 × 7 49 16 × 3 1 6 every square number is a multiple of 4 or exceeds a multiple of 4 by unity ex 5 × 5 25 6 × 4 1 7 × 7 49 12 × 4 1 7 if a square numbers ends in `9 the preceding digit is even ex 7 × 7 49 `4 is the preceding even numbers 27 × 27 729 `2 is the preceding even numbers characteristics of square roots of numbers 1 if a square number ends in `9 its square root is a number ending in 3 or `7 2 if a square number ends in `1 its square root is a number ending in 1 or `9 3 if a square number ends in `5 its square root is a number ending in 5 4 if a square number ends in `4 its square root is a number ending in 2 or `8 5 if a square number ends in `6 its square root is a number ending in 4 or `6 6 if a square number ends in `0 its square root is a number ending in `0 ex © entranceguru.com private limited basic concepts.doc -4-

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p. 5

takes you to places where you belong 529 23 i 729 27 1089 33 1369 37 etc 121 11 ii 81 9 961 31 361 19 625 25 iii 1225 35 2025 45 so on 484 22 iv 64 8 1024 32 784 28 so on 196 14 v 256 16 576 24 676 26 so on vi 100 10 400 20 10000 100 so on © entranceguru.com private limited basic concepts.doc -5-

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p. 6

takes you to places where you belong theory of indices 1 am × an am n 2 amn amn am am-n 3 an 4 abm ambm 5 a0 1 6 ap/q qth root of ap q ap 7 a1 p pth root of a amb m ab 8 cm c 9 a 10 a 0 m 1.find the square root of 6561 factor method 3 3 3 3 3 3 3 6561 2187 729 243 81 27 9 3 6561 3×3×3×3×3×3×3×3 9×9×9×9 81×81 6561 81 2 find the least number with which you multiply 882 so that the product may be a perfect square first find the factors of 882 882 2 × 3 × 3 × 7 × 7 now 882 has factors as shown above `3 repeated twice `7 repeated twice and `2 only once so when one more factor `2 is used then it becomes a perfect square 882 × 2 2 × 2 × 3 × 3 × 7 × 7 the least number required is `2 © entranceguru.com private limited basic concepts.doc -6-

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p. 7

takes you to places where you belong 2 882 3 441 3 147 7 49 7 3 find the cube root of 2985984 factor method 2985984 23 × 23 × 23 × 23 × 33 × 33 3 2985984 2 × 2 × 2 × 2 × 3 × 3 16 × 9 144 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 2985984 1492992 746496 373248 186624 93312 46656 23328 11664 5832 2916 1458 729 243 81 27 9 3 3 1 4 find the value of 4 0.04 4 0.04 4 0.04 4 0.04 4 0.2 3.8 38 19 0.9 4 0.2 4.2 42 21 9 90 30 0.3 3 5 81 0.09 81 0.09 6 simplify 4 3 3 4 © entranceguru.com private limited basic concepts.doc -7-

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p. 8

takes you to places where you belong 4 3 3 2 3 4-3 1 4 2 3 2 3 2 3 8 find the value of 410 1 16 410 1 16 6561 16 6561 16 1 81 20 4 4 6.find the least number with which you multiply 882 so that the product may be a perfect square first find the 2 3 3 7 factors of 882 882 441 147 49 7 now 882 has factors as shown above `3 repeated twice `7 repeated twice and `2 only once so when one more factor `2 is used then it becomes a perfect square 882×2 2×2×3×3×7×7 the least number required is `2 7 find the cube root of 2985984 factor method 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 2985984 1492992 746496 373248 186624 93312 46656 23328 11664 5832 2916 1458 729 243 81 27 9 3 2985984 23×23×23×23×33×33 © entranceguru.com private limited basic concepts.doc -8-

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p. 9

takes you to places where you belong 3 2985984 2 × 2 × 2 × 2 × 3 × 3 16 × 9 144 4 0.04 4 0.04 4 0.2 3.8 38 19 0.9 approx 4 0.2 4.242 42 21 9 90 30 0.3 3 8.find the value of 4 0.04 4 0.04 9 81 0.09 81 0.09 10 simplify 4 3 3 4 4-3 4 3 2 3 1 3 4 2 2 3 3 2 3 11 find the least number with which 1728 may be added so that the resulting number is a perfect square 42 4 1728 16 82 128 164 note take the square root of 1728 by long division method it comes to 41 something as shown if 128 is made 164 we get the square root as an integer the difference between 164 and 128 i.e 36 must be added to 1728 so that 1764 is a perfect square 1764 =42 theory of indices problems 1 a certain number of persons agree to subscribe as many rupees each as there are subscribers subscriber let the number of subscribers be x say since each subscriber agrees to subscribe x rupees © entranceguru.com private limited basic concepts.doc -9the whole subscription is rs.2582449 find the number of

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p. 10

takes you to places where you belong the total subscription no of persons × subscription per person x × x x2 given x2 2582449 x 1607 2 simplify 3 192a 3 b 4 use the 2 formulas abcm a amb mcmmn amn 3 192a3b 4 192a3b 4 1 3 1 3 1 4 3 b 2 × 3 .ab 192 3 a3 6 1 4 3 1 4 2 2.3 3 .ab 3 4a 3b 4 3 1 3 4a 3b 4 3 simplify sol 3 3 x 9 y 12 x 9 y 12 x 9 y 12 1 3 x9 y 1 3 1 12 3 x3y 4 4 find the number whose square is equal to the difference between the squares of 75.12 and 60.12 sol let `x be the number required x2 75.122 ­ 60.122 75.12 60.12 75.12 ­ 60.12 135.24 ×15 2028.60 x 2028 .60 45.0399 © entranceguru.com private limited basic concepts.doc 10 -

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p. 11

takes you to places where you belong 5 reduce to an equivalent fraction write rational denominator sol 3 5 3 5 3 3 5 3 3 3 5 3 5 3 15 3 15 15 3 5-3 18 4 15 9 2 15 2 6 find the value of sol 14 14 7 9 21 ×2 × 55 10 44 21 7 9 ×2 × 44 55 10 637 117 9 × × 5.26 approx 44 55 10 7 an army general trying to draw men in the front row his 16,160 men in rows so that there are as many men as true are rows found that he had 31 men over find the number of let `a be the number of men in the front row a2 31 161610 no of men in the front row 127 a2 161610 ­ 31 16129 a 127 8 a man plants his orchid with 5625 trees and arranges them so that there are as many rows as there are trees in a row how many rows are there sol let `x be the number of rows and let the number of trees in a row be `x say x 5625 x 75 there are 75 trees in a row and 75 rows are arranged 2 © entranceguru.com private limited basic concepts.doc 11 -

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p. 12

takes you to places where you belong time and work points for recapitulation while solving problems on time and work 1 if a person can do a piece of work in `m days he can do 2 if the number of persons engaged to do a piece of work be increased or decreased in a certain ratio the time required to do the same work will be decreased or increased in the same ratio 3 if a is twice as good a workman as b then a will take half the time taken by b to do a certain piece of work 4 time and work are always in direct proportion 5 if two taps or pipes p and q take `m and `n hours respectively to fill a cistern or 1 1 tank then the two pipes together fill part of the tank in 1 hour and the m n entire tank is filled in 1 of the work in 1 day m 1 mn hours 1 1 m+n m n examples 1 if 12 man can do a piece of work in 36 days in how many days 18 men can do the same work solution 12 men can do a work in 36 days 12 18 men can do the work in × 36 24 days 18 note if the number of men is increased the number of days to finish the work will decrease 2 a and b can finish a work in 12 days b and c can finish the same work in 18 days c and a can finish in 24 days how many days will take for a b and c combined together to finish the same amount of work solution a and b can finish the work in 12 days 1 a+b can finish in 1 day of the work 12 1 of the work similarly b+c can finish in 18 1 of the work c+a can finish in 24 1 1 1 2a+b+c can finish in 1 day of the work 12 18 24 6 4 3 13 of the work 72 72 13 a+b+c can finish in 1 day of the work 144 © entranceguru.com private limited time and work 5 pages doc -1-

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p. 13

takes you to places where you belong 144 1 11 days 13 13 3 a b and c earn rs.120 per day while a and c earn rs.80 per day and b and c earn rs.66 per day find c s earning only solution a+b+c earn per day rs.120 1 a+c earn per day rs.80 2 b+c earn per day rs.66 3 from 1 and 2 we find that c earns rs.40 from 3 we get that b earns 26 since c earns rs.40 4 4 men or 8 women can do a piece of work in 24 days in how many days will 12 men and 8 women do the same work solution a+b+c can together finish the work in 4 men 8 women can do a work in 24 days 1 1 man or 2 women in one day can finish of the work 24 now 12 men and 8 women together will do the work as 16 men alone can do the work 16 2 16 men can do in one day of the work 24 3 1 3 or 1 days 16 men can do the entire work in 2 2 5 a and b can finish a work in 16 days while a alone can do the same work in 24 days in how many days b alone can finish the same work solution a and b can finish a work in 16 days 1 of the work a+b can finish in one day 16 1 of the work a alone can finish in one day 24 the amount of work that b alone can do in one day 1 1 3-2 1 of the work 16 24 48 48 b alone can complete the work in 48 days 6 a and b can do a piece of work in 12 days b and c can do it in 20 days if a is twice as good a workman as c then in what time will b alone do it solution a+b can do a work in 12 days 1 a+b in one day can do of the work 1 12 1 of the work 2 similarly b+c in one day can do 20 since a=2c in 1 put a=2c 1 2c+b in one day can do of the work 12 © entranceguru.com private limited time and work 5 pages doc -2-

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p. 14

takes you to places where you belong i.e b+c c in one day can do c alone can do in one day 1 of the work 12 5-3 2 1 1 1 of work 3 60 60 30 12 20 1 1 3-2 1 from 2 using 3 b alone can do in one day of work 20 30 60 60 b alone can do the entire work in 60 days 7 a contractor undertook to do a piece of work in 125 days and employs 175 men to carry out the job but after 40 days he finds that one a quarter of the work had been carried out how many more men should be employed to finish the work in time solution men days work 1 175 40 4 3 x 85 4 3 40 x 175 × × 4 1 85 4 247 247 ­ 175 or 72 men will have to be employed 8 b is twice as fast as a and c is twice as fast as b if a alone can complete the job in 28 days how many days will a b and c take to complete the job working together solution given b 2a c 2b efficiency-wise a can complete the work in 28 days given b alone can complete the work in 14 days [b is twice as fast as a hence c alone can complete the work in 7 days [c is twice as fast as b 1 1 1 a+b+c working together can complete part of the 28 14 7 work in 1 day 1+2+4 7 1 of the work in one day 28 28 4 a+b+c working together can complete the entire work in 4 days 9 a cistern is normally filled in 6 hours but takes 2 hours more to fill it because of a leak at its bottom if the cistern is full how long will it take for the leak to empty the cistern solution © entranceguru.com private limited time and work 5 pages doc -3-

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p. 15

takes you to places where you belong the cistern is filled in 6 hours 1 in 1 hour of the cistern will be filled 6 1 of the cistern only is filled due to a leak in the cistern in 1 hour 8 1 1 4-3 1 leakage in 1 hour of the cistern 6 8 24 24 1 of the cistern is leaked out in 1 hour 24 the entire cistern will become empty in 24 hours 10 two pipes a and b are attached to a cistern pipe a can fill the cistern in 6 hours while pipe b can empty it in 8 hours when both the pipes are opened together find the time for filling the cistern solution 1 pipe a fills in an hour of the cistern 6 1 of the cistern pipe b empties in one hour 8 1 1 1 in 1 hour the portion of the cistern filled 6 8 24 time taken to fill the cistern 24 hours 11 the efficiency of the first machine tool is 20 less than that of the 2nd one the first machine tool operated for 5 hours whereas the second one operated for 4 hours and together they machined 4000 work pieces find the number of work-pieces machined by the first machine tool solution efficiency of the two machine tools 80 100 4 5 time ratio of the two machine tools 5 4 work ratio of the two machine tools 4×5 5×4 20 20 1 1 thus both the machine tools produced the same number of workpieces 4000 viz 2000 2 alternative the second machine operating for 4 hours first machine operating for 5 hours hence the first machine would produce 4000 pieces in 10 hours in 5 hours it would have produced 2000 pieces 12 two coal loading trucks handle 9000 tonnes of coal at an efficiency of 90 working 12 hours per day for 8 days how many hours a day 3 coal loading trucks should work at an efficiency of 80 so as to load 12000 tonnes of coal in 6 days solution © entranceguru.com private limited time and work 5 pages doc -4-

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