hcf and lcm solving methods, Exercises of Engineering

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Quantitative Aptitude

Concept 1

1. Number System

2. HCF and LCM

Prelims Paper II

 In Hindu Arabic System, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called digits to represent any number. This is the decimal system where we use the numbers 0 to 9.  0 is called insignificant digit.

 Natural numbers  Counting numbers 1, 2, 3, 4, 5.  The set of all natural numbers can be represented by

 Whole numbers  If we include 0 among the natural numbers, then the numbers 0, 1, 2, 3, 4, 5 … are called whole numbers.  The set of whole number can  Every natural number is a whole number but 0 is a whole number which is not a natural number.

 Integers  All counting numbers and their negatives including zero are known as integers.  The set of integers can be repre

• • • • • • • • • • • • •

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NUMBER SYSTEM

NUMBER SYSTEM

In Hindu Arabic System, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called digits to represent any number. This is the decimal system where we use the numbers 0 to 9. 0 is called insignificant digit.

Counting numbers 1, 2, 3, 4, 5... are known as natural numbers. The set of all natural numbers can be represented by N= {1, 2, 3, 4, 5…}

If we include 0 among the natural numbers, then the numbers 0, 1, 2, 3, 4, 5 … are called whole numbers. The set of whole number can be represented by W= {0, 1, 2, 3, 4, 5…} Every natural number is a whole number but 0 is a whole number which is not a

All counting numbers and their negatives including zero are known as integers. The set of integers can be represented by Z or I = {…-4, -3, -2, -1, 0, 1, 2, 3, 4 …}

Types of Numbers

• Natural numbers
• Whole numbers
• Integers
• Positive Integers
• Negative Integers
• Non-negative Integers
• Rational Numbers
• Irrational Numbers
• Real Numbers
• Even Numbers
• Odd Numbers
• Prime Numbers
• Composite Numbers

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NUMBER SYSTEM 2

In Hindu Arabic System, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called digits to represent

N= {1, 2, 3, 4, 5…}

If we include 0 among the natural numbers, then the numbers 0, 1, 2, 3, 4, 5 … are

Every natural number is a whole number but 0 is a whole number which is not a

All counting numbers and their negatives including zero are known as integers. 1, 0, 1, 2, 3, 4 …}

 Even Numbers  All those numbers 8, 10, etc., are even numbers.

 Odd Numbers  All those numbers which are not exactly divisible by 2 are called odd numbers, e.g. 1, 3, 5, 7 etc., are odd numbers.

 Prime Numbers  A natural number other than 1 is a prime number if it is divisible by 1 and itself only.  For example, each of the numbers 2, 3, 5, 7 etc., are prime numbers.

 Composite Numbers  Natural numbers greater than 1which are not prime, are known as composite numbers.  For example, each of the numbers 4, 6, 8, 9, 12, etc., are composite numbers.

The number 1 is neither a prime number nor composite number.

2 is the only even number which is prime

Prime numbers up to 100 are: 2, 3, 5, 7, 11, 13, 17, 19,23, 29, 31, 37, 41,43, 47, 53, 59,

61, 67, 71, 73, 79, 83, 89, 97, i.e. 25 prime numbers between 1 and 100.

Two numbers which have only 1 as the common factor are called co

relatively prime to each other, e.g. 3 and 5 are co

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NUMBER SYSTEM

which are exactly divisible by 2 are called even numbers, e.g.2, 6, 8, 10, etc., are even numbers.

All those numbers which are not exactly divisible by 2 are called odd numbers, e.g. 1, 3, 5, 7 etc., are odd numbers.

umber other than 1 is a prime number if it is divisible by 1 and itself only. For example, each of the numbers 2, 3, 5, 7 etc., are prime numbers.

Natural numbers greater than 1which are not prime, are known as composite example, each of the numbers 4, 6, 8, 9, 12, etc., are composite numbers.

BITS

The number 1 is neither a prime number nor composite number.

2 is the only even number which is prime

Prime numbers up to 100 are: 2, 3, 5, 7, 11, 13, 17, 19,23, 29, 31, 37, 41,43, 47, 53, 59,

61, 67, 71, 73, 79, 83, 89, 97, i.e. 25 prime numbers between 1 and 100.

Two numbers which have only 1 as the common factor are called co-primes or

relatively prime to each other, e.g. 3 and 5 are co-primes.

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NUMBER SYSTEM 4

which are exactly divisible by 2 are called even numbers, e.g.2, 6,

All those numbers which are not exactly divisible by 2 are called odd numbers, e.g. 1,

umber other than 1 is a prime number if it is divisible by 1 and itself only.

Natural numbers greater than 1which are not prime, are known as composite example, each of the numbers 4, 6, 8, 9, 12, etc., are composite numbers.

Prime numbers up to 100 are: 2, 3, 5, 7, 11, 13, 17, 19,23, 29, 31, 37, 41,43, 47, 53, 59,

primes or

H.C.F. and L.C.M.

Common factor

A common factor of two or more numbers is a number which divides each of them exactly. For example, 4 is a common factor of 8 and 12.

Highest common factor (H.C.F.) Highest common factor of two or more numbers is the greatest number that divides each one of them exactly. For example, 6 is the highest common factor of 12, 18 and 24. Highest Common Factor is also called Greatest Common Divisor or Greatest Common Measure. Symbolically, these can be written as H.C.F. or G.C.D. or G.C.M., respectively.

Methods of Finding H.C.F.

I. Method of Prime Factors

Step 1 Express each one of the given numbers as the product of prime factors. [A number is said to be a prime number if it is exactly divisible by 1 and itself but not by any other number, e.g. 2, 3, 5, 7, etc. are prime numbers]

Step 2 Choose Common Factors.

Step 3 Find the product of lowest powers of the common factors. This is the required H.C.F. of given numbers.

 H.C.F. = 2^3  3 5  5 9.

II. Method of Division

A. For two numbers:

Step 1 Greater number is divided by the smaller one. Step 2 Divisor of (1) is divided by its remainder. Step 3 Divisor of (2) is divided by its remainder. This is continued until no remainder is left. H.C.F. is the divisor of last step.

Example 5 Find the H.C.F. of 3556 and 3444. Solution 3444)3556 ( 3444________

112. 3444 ( 3360______
113. 112 ( 84________
114. 84 ( 84  HCF=Divisor of last step=

B. For more than two numbers: Step 1 Any two numbers are chosen and their H.C.F. is obtained. Step 2 H.C.F. of H.C.F. (i.e. HCF obtained in step 1) and any other number is obtained. Step 3 H.C.F. of H.C.F. (i.e. HCF obtained in last step) and any other number (not chosen earlier) is obtained. This process is continued until all numbers have been chosen. H.C.F. of last step is the required H.C.F.

Example 6 Find the greatest possible length which can be used to measure exactly the lengths 7 m, 3 m 85 cm and 12 m 95 cm Solution Required length = HCF of 7 m, 3 m 85 cm and 12 m 95 cm =HCF of 700 cm, 385 cm and 1295 cm = 35 cm.

Common Multiple

A common multiple of two or more numbers is a number which is exactly divisible by each one of them.

For Example, 32 is a common multiple of 8 and 16. 8  4 = 32 16  2 = 32

320 = 2x2x2x2x2x2x  L.C.M. = 2x2x2x2x2x2x3x 5 = 960.

II. Method of Division

Step 1 The given numbers are written in a line separated by common. Step 2 Divide by any one of the prime numbers 2, 3, 5, 7, 11 … which will divide at least any two of the given nu8mbers exactly. The quotients and the undivided numbers are written in a line below the first. Step 3 Step 2 is repeated until a line of numbers (prime to each other) appears. Find the product of all divisors and numbers in the last line which is the required L.C.M.

Example 8 Find the L.C.M. of 12, 15, 20 and 54.

Solution 2 12 15 20 54 2 6 15 10 27 3 3 15 5 27 3 1 5 1 9 3 1 5 1 3 5 1 5 1 1 1 1 1 1 L.C.M. = 2  2  3  3  3  5 = 540

Note: Before finding the L.C.M. or H.C.F., we must ensure that all quantities are expressed in the same unit.

Tips

1. H.C.F. and L.C.M. of Decimals Make the same number of decimal places in all the given numbers by suffixing zero(s) if necessary. Find the H.C.F. /L.C.M. of these numbers without decimal. Put the decimal point (in the H.C.F. /L.C.M. of step 2) leaving as many digits on its right as there are in each of the numbers.
2. L.C.M. and H.C.F. of Fractions

ܯ .ܥ .ܮ. =ݐ݂݋ .ܨ .ܥ .ܪݐ݂݋. ܯ .ܥ .ܮ݊݁݀ ݊݅ ݏݎܾ݁݉ݑ݊ ݁ℎ݊ ݊݅ ݏݎܾ݁݉ݑ݊ ݁ℎܽ݊݅݉݋ݐܽݎ݁݉ݑݏݎ݋ݐݏݎ݋

= .ܨ .ܥ .ܪݐ݂݋. ܯ .ܥ .ܮݐ݂݋ .ܨ .ܥ .ܪ݊݁݀ ݊݅ ݏݎܾ݁݉ݑ݊ ݁ℎ݊ ݊݅ ݏݎܾ݁݉ݑ݊ ݁ℎܽ݊݅݉݋ݐܽݎ݁݉ݑݏݎ݋ݐݏݎ݋

3. Product of two numbers = L.C.M. of the numbers  H.C.F. of the numbers
4. To find the greatest number that will exactly divide x, y and z. Required number = H.C.F. of x, y and z
5. To find the greatest number that will divide x, y and z leaving remainders a, b and c , respectively. Required number = H.C.F. of ( x – a ), ( y – b ) and ( z – c )
6. To find the least number which is exactly divisible by x, y and z. Required number = L.C.M. of x, y and z
7. To find the least number which when divided by x, y and z leaves the remainders a, b and c, respectively. It is always observed that (x – a) = (y – b) = (z – c) = k (say)  Required number = (L.C.M. of x, y and z ) – k.