QUANTITATIVE APTITUDE

Transcription

1 QUANTITATIVE APTITUDE HCF AND LCM Important Points : Factors : The numbers which exactly divide a given number are called the factors of that number. For example, factors of 15 are 1, 3, 5 and 15. Common Factor: When a unique number divides exactly two or more given numbers then that number is called the common factor of the given For example, each of the numbers 2, 3 and 6 is a common factor of 12 and 18. Multiple : When a number is exactly divisible by another number, then the former number is called the multiple of the latter number. Obviously, latter number is contained in the former. For example, 24 is a multiple of 1, 2, 3, 4, 6, 8, 12 and 24. Prime Number: A number is said to be a prime number if its factors are 1 and the number itself. For example, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, are prime Co-prime : Two numbers are said to be co-prime (prime to each other) if there is no common factor other than 1 between them. It should be noted that the numbers which are co-prime, are not necessarily prime Even composite numbers may be mutually prime. For example, 5 and 7 are co-prime. Again, 16 and 25 are also co-prime where neither of them is a prime number. Highest Common Factor : The highest common factor (HCF) also known as greatest common divisor (GCD) or greatest common measure (GCM) of two or more numbers is the greatest number which divides each of them exactly. The HCF of two co-prime numbers is 1. It can be taken as a test to decide whether two given numbers are co-prime or not. For example, Let's consider two numbers 24 and 36. All possible factors of 24 are 1,2,3, 4, 6, 8, 12 and 24 All possible factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36. The common factors of 24 and 36 are 1, 2, 3, 4, 6 and 12 The greatest common factor is 12. Hence, 12 is the HCF of 24 and 36. METHODS TO FIND OUT HCF : There are two methods of determining the HCF of two or more (i) Method of Prime Factors : First of all resolve the given numbers into prime factors and then find the product of all the prime factors common to all the given The product is the required HCF. - Example : Find the HCF of 126, 396 and 5400 Sol.: 126 = 2x3x3x7 396 = 2x2x3x3x = 2x2x2x3x3x3x5x5 Common factors are 2, 3 and HCF= 2 x 3 x 3 = 18 (il) Method of Division : Suppose we are given two numbers whose HCF is to be determined. Divide the greater number by the smaller one. Now, divide the divisor by the remainder, divide the remainder by the next remainder and so on, until'no remainder (o-remainder) is left. The last divisor 50 obtained is the required HCF of the two given If there are three or more numbers whose HCF is to be determined, then first find the HCF of any two of the three numbers and then find the HCF of the third number and the HCF of these two numbers and so on. The last HCF will be the required HCF. Example : Find the HCF of 48, 168 and 324 Sol. : Firstly, we find the HCF of 48 and HCF of 48, 168 and 324 is 12. Lowest Common Multiple : The lowest common multiple (LCM) of two or more given numbers is the least number which is exactly divisible by each of them. For example, Multiples of 12 are 12, 24, 36, 48, 60, 72 Multiples of 18 are 18, 36, 54, 72,... Common multiples are 36, LCM = 36 METHOD TO FIND LCM : Put the numbers side by side putting comma in a row. Divide by any of the prime numbers 2, 3, 5, 7 which divides at least any two of the given numbers exactly. Write down the quotients so obtained and the indivisible numbers side by side and proceed in the way until you get a row of numbers which are prime to one another. The continued product of all the divisors and the numbers in the last row will be the required LCM.

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3 4. Find the least number of soldiers in a regiment which can allow it to be drawn up in a hollow square 8, 10, 12 or 15 deep and also into a solid square. Sol. : Number of soldiers required to be drawn up into hollow square of 8, 10, 12 or 15 deep = 4 x (LCM of 8, 10, 12, 15) because they stand only on the perimeter. 2 8, 10, 12, , 5, 6, , 5, 3, ; 5, 1, 5 2, 1, 1, 1

4 EVALUATE YOURSELF SUBJECTIVE TYPE QUESTIONS 1. Four drums containing water have capacities 135 litres, 205 litres, 165 litres and 240 litres. Find the greatest capacity measure of the drums. 2. Find the least multiple of 13 which when divided by 6, 8 and 12 leaves 5, 7 and 11 as remainders respectively. 3. The ratio between the LCM and HCF of two numbers is 28 : 1. If the numbers be 96 and 168, find their LCM and HCF. 4. A labourer was engaged for a certain number of days for Rs , but being absent on some of those days he was paid only Rs. 50. Prove that his daily wages could not be more than Rs In finding the GCM of two numbers the last remainder is 35 and the quotients are 1,2, 1,3. Find the 6. Find the least number which is divisible by all the numbers from 1 to 20 inclusive. 7. Three men start together to travel the same way around a circular track of 11 kms in circumference. Their speeds are 4, 5% and 8 kms per hour respectively. When will they meet at a starting point? 8. The circumference of the wheels of a carriage are 2m 4dm 5 cm and 3m 5 dm, what is the least distance in which both the wheels will make an exact number of revolutions? 9. What is the least number that must be added and what is the greatest number that must be subtracted from that the results may be divisible by 777, 819 and 4329? 10. A heap of pebbles can be made up exact ly into groups of 25, but when made up into groups of 18, 27 and 32, there is in each case a remainder of 11; find the least number of pebbles such a heap can contain. 11. An electronic device makes a beep every 60 sec. Another device makes a beep after every 62 sec. They beeped together at 10 a.m. Find the time when they will next make a beep together at the earliest. 12. The traffic lights at three different road crossings change after every 48 sec, 72 sec. and 108 sec. respectively. If they all change simultaneously at 8 : 30 : 00 hours then they will again change simultaneously at what time? 13. The HCF and LCM of two numbers are respectively 12 and 2448 and their difference is 60. Find the 14. Two cog wheels revolve together. One of the wheels has 24 teeth and the other has 30 teeth. The smaller one makes 200 revolutions in 1 min. Find the number of revolutions made by larger wheel in an hour. 15. A gardener was asked to plant flowers in a row containing equal number of plants. He tried to plant 6, 8, 10 and 12 in each row, but 5 plants left in each case but when he planted 13 in a row, no plant was left. Find the least no. of plants with him. OBJECTIVE TYPE QUESTIONS 16. The H.C.F. and L.C.M. of two numbers are 44 and 264 respectively. If the first number is divided by 2. the quotient is 44. The other number is (1)33 (2) 66 (3) 132 (4) The traffic lights at three different road crossings change after every 48 sec, 72 sec. and 108 sec. respectively. If they all change simultaneously at 8 : 20 : 00 hrs; then they will again change simultaneously at (1) 8 : 27.: 12 hrs. (2) 8 : 27 : 24 hrs. (3) 8 : 27 : 36 hrs. (4) 8 : 27 : 48 hrs. 18. What least number must be subtracted from 1294 so that the remainder when divided by 9, 11, 13 will leave in each case the same remainder 6? (1)0 (2) 1 (3) 2 (4) The largest natural number which exactly divides the product of any four consecutive natural numbers is (1)6 (2) 12 (3)24 (4) Six bells" commence tolling together and toll at intervals of 2, 4, 6, 8, 10 and 12 seconds respectively. In 30 minutes, how many times do they toll together? (1)4 (2) 10 (3) 15 (4) 16 (1)45 (2) 35 (3)30 (4) About the number of pairs which have 16 as their H.C.F. and 136 as their L.C.M., we can deninitely say that

5 (1) Only one such pair exists (2) Only two such pairs exist (3) Many such pairs exist (4) No such pair exists 23. A worker was engaged for a certain number of days and was promised to be paid Rs, He remained absent for some days and was paid Rs only. What were his daily wages? (1) Rs. 27 (2) Rs. 28 (3) Rs. 29 (4) Rs The H.C.F. and L.C.M. of a pair of numbers are 12 are 924 respectively. How many such distinct pairs are possible? il) 1 (2) 2 (3) 3 (4) If m = and n = then the H.C.F. of m. and n is given by (1) (2) (3) (4) A man was employed on the promise that he will be paid the highest wages per day. The contract money to he paid was Rs Finally he was paid only Rs For how-many days did he actually work? (1) 39 (2) 40 (3)37 (4) An officer was appointed on maximum daily wages on a contract money of Rs But on being absent for some days, he was paid only Rs For how many days was he absent? (l) 3 (2) 4 tap S (4) e 28. A merchant has three kinds of milk : 435 litres, 493 litres and 551 litres. Find the least number of casks of equal size required to store all the milk without mixing. (1) 51 (2) 61 (3)47 (4) A wholesale tea dealer has 408 kg. 468 kg and 516 kg of three different qualities of tea. He wants it all to be packed into boxes of equal size without mixing. Find the capacity of the largest possible box. (1) 50 (2) 36 (3) 24 (4) What is the least number of cut pieces of equal length that can be cut out of two lengths 10 metres 857 milllimetres and 15 metres 87 millimetres. (1) 174 (2) 172 (3) 164 (4) 184 ANSWERS 1. 5 litres and and hours m 50 cm 9. 9 and am : 37 : 48 hours and (3) 17. (1) 18. (2) 19. (3) 20. (4) 21.(3) 22.(4) 23. (3) 24.(2) 25.(1) 26.(3) 27.(1) 28. (1) 29. (4) 30. (4)