Lakshesh Verma Lakshesh Verma

Lecture 1: Euclid’s division Lemma & The Fundamental theorem of arithmetic

  • Real Numbers Assignments 1

    Show that one and only one out of n,n+2,n+4 is divisible by 3, where n is any positive integer.

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  • Real Numbers Assignments 2

    The sets of mathematics, physics and physical education books have to be stacked
    in such a way that all the books are stored topic wise. The number of mathematics
    , physics and physical education books are 14 , 18 and 22. Determine the number
    of stacks of each books provided books are of the same thickness.

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  • Real Numbers Assignments 3

    In a morning walk Nirmaljeet, Puneet, Rajiv step off together, their steps
    measuring 240 cm , 90 cm, 120cm respectively. What is the minimum distance
    each should walk so that one can cover the distance in complete steps?

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  • Real Numbers Assignments 4

    Find the largest positive integer that will divide 122,150 and 115 leaving
    remainder 5, 7 and 11 respectively.

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  • Real Numbers Assignments 5

    Use Euclid‟s division algorithm to find the largest number which divides 957 and
    1280 leaving remainder 5 in each case.

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  • Real Numbers Assignments 6

    Find the HCF of 616 and 32 using Euclid’s division algorithm. 

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  • Real Numbers Assignments 7

    Show that the square of any positive integer is of the form 3m or 3m+1,for some
    integer m.

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  • Real Numbers Assignments 8

    Find the LCM and HCF of 12 , 15 and 21 by prime factorization method.

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  • Real Numbers Assignments 9

    The HCF of two number is 4 and their LCM is 9696,if one number is 96,find the
    other number.

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  • Real Numbers Assignments 10

    Find the LCM and HCF of 91 and 26, also verify that HCF×LCM= product of
    two

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  • Real Numbers Assignments 11

     Find HCF of 128 and 216. 

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  • Real Numbers Assignments 12

    There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

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  • Real Numbers Assignments 13

     

    Check whether 6n can end with the digit 0 for any natural number n.

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  • Real Numbers Assignments 14

    Given that HCF (306, 657) = 9, find LCM (306, 657).

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  • Real Numbers Assignments 15

      

    Find the LCM and HCF of the following integers by applying the prime factorisation method.

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  • Real Numbers Assignments 16

     

    Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.

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  • Real Numbers Assignments 17

    Express each number as product of its prime factors:

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  • Real Numbers Assignments 18


    Use Euclid's division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m +8.

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  • Real Numbers Assignments 19

      

    Use Euclid's division lemma to show that the square of any positive integer is either of form 3m or 3m + 1 for some integer m.

    [Hint: Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square each of these and show that they can be rewritten in the form 3m or 3m + 1.]

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  • Real Numbers Assignments 20

     

    An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

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  • Real Numbers Assignments 21

    Show that any positive odd integer is of the form , or , or , where q is some integer.

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  • Real Numbers Assignments 22

     

    Use Euclid's division algorithm to find the HCF of:

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    • Description:-

    Lecture 1:
    Today, we will discuss the first two topics, i.e; Euclid’s division Lemma & The Fundamental theorem of arithmetic respectively of Chapter 1 Real Numbers
    Objectives:
    1. Basic concept about Euclid division Lemma.
    2. Concept of finding HCF & LCM.
    Important Tip: 
    This lecture is about Euclid division Lemma and Fundamental theorem of arithmetic. These topics help in many ways as you can learn finding HCF by Euclid division Lemma & LCM by prime factorization method. Have a clear focus on these topics as it help you in further studies also.
    Please go through the topic, their notes. Properly do the assignments, watch videos of the concerned topics. If you have any doubt about any topic/point please submit it in the discussion section... it will be answered as soon as possible.
    After all of this... when you feel confident about the particular topic/chapter then take the quiz. 

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