## NUMBER THEORY

### Introduction

Highest common factor :

If there are three numbers like 10, 20, 30 and there exists numbers that can divide them to give remainder 0 viz. 5, 10. Then the lowest of those numbers is the HCF viz. 5

Least Common multiple:

If we take 4,6 as two numbers then both of them divide 12, 24, 36 etc fully giving remainder 0 so the lowest such number common to both is the LCM which in this case is 12.

### Questions:

Q. What is the least number of cans needs for a vendor who has 21 L of Milk ‘a’ , 42 L of milk ‘b’ and 63L of Milk ‘c’. If all cans should have same number of milk and he should need least number of cans.

Ans: Find HCF of 21, 42, 63 which is 21. So total cans are 6 i.e. 1 for milk ‘a’ , 2 for milk ‘b’ and 3 for milk ‘c’.

Q. If the HCF and LCM of two numbers is 18, 180 and one of the number is 36. Then the second number is.

Ans. HCF * LCM = a * but HCF = 18, LCM = 180 and a= 36 so substituting we get b = 90.

### Formulae of HCF and LCM

• If a,b are consecutive even numbers their LCM is a*b/2;
• HCF of fractions = HCF of numerator / LCM of denominator;
• LCM of fractions = LCM of numerators / HCF of denominators;
• Number of zeroes at the end of the factorial = n! [ n/5 ] + [ n/5^2 ] + [ n/5^3 ] ….
• If a number “a” divides another number “b” exactly, we say that “a” is a factor of “b”. In this case, “b” is called a multiple of “a”.
• Suppose we have to find the H.C.F. of three numbers. Then, H.C.F. of [(H.C.F. of any two) and (the third number)] gives the H.C.F. of three given numbers.
• Product of two numbers = Product of their H.C.F. and L.C.M
• Co-primes: Two numbers are said to be co-primes if their H.C.F. is 1
• H.C.F. and L.C.M. of Fractions :
• HCF = H.C.F. of NumeratorsL.C.M. of Denominators
• LCM = L.C.M. of NumeratorsH.C.F. of Denominators
• H.C.F. and L.C.M. of Decimal Fractions : In given numbers, make the same number of decimal places by annexing zeros in some numbers, if necessary. Considering these numbers without decimal point, find H.C.F. or L.C.M. as the case may be. Now, in the result, mark off as many decimal places as are there in each of the given numbers.
• HCF of two consecutive even numbers is 2. HCF of two consecutive odd numbers is 1.
• If the HCF of x and y is G, then the HCF of x, (x + y) is also G. x, (x – y) is also G
• HCF of (am−1) and (an−1) is given by (aHCF of m,n−1)

### Strategy for Finding LCM

• Rules of Co-prime numbers :

1.Two consecutive natural numbers are always co-prime

2.Two consecutive odd numbers are always co-prime

3.Two prime numbers are always co-prime

4.One prime number and another composite number (such that the composite number is not a multiple of the prime number) are always co-prime (Examples: 17, 38; 23, 49 and so on, but note that 17 and 51 are not co-prime)

5.Three or more numbers being co-prime with each other means that all possible pairs of the numbers would be co-prime with each other. Thus, 47, 49, 51 and 52 are co-prime since each of the 6 pairs (47,49); (47,51); (47,52); (49,51); (49,52) and (51,52) are co-prime.

6.Three consecutive odd numbers are always co-prime

7.Three consecutive natural numbers with the first one being odd are coprime

8.Two consecutive natural numbers along-with the next odd number such that the first no. is even (examples: 22, 23, 25; 52, 53, 55; 68, 69, 71 and so on)

9.Three prime numbers (Examples: 17, 23, 29; 13, 31, 43 and so on)

• Let us say that you were trying to find the LCM of 9,10,12 and 15. The rules of Co-prime numbers are beneficial for you !

1.Step 1 : If you can see a set of 2 or more co-prime numbers in the set of numbers for which you are finding the LCM write them down by multiplying them. So in the above situation, since we can see that 9 and 10 are co-prime to each other we can start off writing the LCM by writing 9 * 10 as the first step.

2.Step 2 : For each of the other numbers, consider what part of them have already been taken into the answer and what part remains outside the answer. In case you see any part of the other numbers such that it is not a part of the value of the LCM you are writing such a part would need to be taken into the answer of the LCM.

3.Thought about 12 : 12 is 2 * 2 * 3. 9 * 10 already has a 3 and one 2 in it prime factors. However, the number 12 has two 2’s. This means that one of the two 2’s of the number 12 is still not accounted for in our answer. Hence, we need to modify the LCM by multiplying the existing 9 * 10 by a 2. With this change the LCM now becomes: 9 * 10 * 2

4.Thought bout 15 : 15 is 5 * 3, however the term 9 * 10 * 2 already has 5 and 3 . Hence there is no need for additionally having a 5 * 3 in the LCM. Hence, the LCM is 9 * 10 * 2.

### Solved Questions

Q. Find the HCF of (3125 – 1) and (335 – 1).

• 325 – 1
• 35 – 1
• 3125 – 1
• 335 – 1

Q. The smallest square number, which is exactly divisible by 2, 3, 4, – 9, 6, 18, 36 and 60, is

• 900
• 1600
• 3600
• none

Q. HCF of 23∗33∗5∗74,22∗35∗52∗73,23∗53∗72

• 1080
• 970
• 960
• 980

Q. If two given numbers are divisible by a number, then their difference is also divisible by that number.

• true
• false
• cant say
• only if numbers are not consecutive

Q. Two tankers contain 850 litres and 680 litres of kerosene oil respectively. Find the maximum capacity of a container which can measure the kerosene oil of both the tankers when used an exact number of times

• 170
• 470
• 370
• 270