Finding Remainders by Simplifying the Dividend - Ravi Handa


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    Are you struggling with Remainders?
    Here is a bunch of questions with detail solutions.

    Type : Simplifying the dividend (Direct)

    What's the remainder when 2^99 is divided by 33?

    To solve these kind of questions, it is important that we modify the dividend. We should break it in such a format that it gets close to the divisor - preferably +1 or -1 of the divisor. For example, in this question we are dividing the numerator by 33. We should try to make the numerator 32 or 34. Since it is powers of 2, making it 32 is possible.

    Let's have a look at the complete and detailed solution below.

    Rem [2^99 / 33]
    = Rem [2^4 * 2^95 / 33]
    = Rem [16 * 32^19 / 33]
    = Rem [16 * (-1)^19 / 33]
    = Rem [ (-16) / 33]
    = 17

    What is the remainder when 17^200 is divided by 18?

    These type of questions become really simple if you understand the concept of negative remainders. Always try and reduce the dividend to 1 or -1.
    Rem [17^200 / 18]
    = Rem [ (-1)^200 / 18]
    = Rem [1 / 18]
    = 1

    What is the remainder when 30^100 is divided by 17?

    Rem [30^100 / 17]
    = Rem[(-4)^100 / 17]
    = Rem[16^50/ 17]
    = Rem[(-1)^50 / 17]
    = Rem[1^50 / 17]
    = 1

    How do you find the remainder of 54^124 divided by 17?

    Rem [54^124 / 17]
    = Rem[(3)^124 / 17]
    = Rem[81^31 / 17]
    = Rem[(-4)^31 / 17]
    = Rem[(-4)^30 * (-4) / 17]
    = Rem[(16)^15 * (-4) / 17]
    = Rem[(-1)^15 * (-4) / 17]
    = Rem[(-1) * (-4) / 17]
    = 4

    How do you find the remainder of 21^875 divided by 17?

    Rem [21^875 / 17]
    = Rem [4^875 / 17]
    = Rem[4 * 4^874 / 17]
    = Rem [4 * 16^437 / 17]
    = Rem [4 * (-1)^437 / 17]
    = Rem [4 * (-1) / 17]
    = Rem [-4 / 17]
    = 13

    What will be the remainder when (16^27+37) is divided by 17?

    Rem [(16^27+37)/17]
    = Rem [16^27/17] + Rem [37/17]
    = Rem [(-1)^27/17] + 3
    = -1 + 3
    = 2

    What is the remainder when 2^33 is divided by 27?

    Rem [2^33/27]
    = Rem [32^6 * 8 / 27]
    = Rem [5^6 * 8 / 27]
    = Rem [ 125^2 * 8 / 27]
    = Rem [ (-10)^2 * 8 / 27]
    = Rem [800/27]
    = 17

    What is the remainder of the following question (71^71+71) / (72)?

    Rem [(71^71 + 71)/72]
    = Rem [71^71/72] + Rem [71/72]
    = Rem [(-1)^72] + (-1)
    = (-1) + (-1)
    = -2
    = 70

    How will you find the remainder when 15^2010+16^2011 is divided by 7?

    Here we need to know that:
    Rem[(a + b)/c] = Rem[a/c] + Rem[b/c]
    Rem[(a*b)/c] = Rem[a/c] * Rem[b/c]

    Keeping that in mind:
    Rem[15^2010/7] = Rem[1^2010/7] = 1

    Rem[16^2011/7]
    = Rem[2^2011/7]
    = Rem[2^2010/7] * Rem[2/7]
    = Rem[8^670/7] * 2
    = 1 * 2 = 2
    Rem[(15^2010 + 16^2011)/7] = 1 + 2 = 3

    What is the remainder when 30^40 is divided by 7?

    Rem [30^40 / 7]
    = Rem[2^40 / 7]
    = Rem[2^39 * 2 / 7]
    = Rem[8^13 * 2 / 7]
    = Rem[1^13 * 2 / 7]
    = Rem[1*2 / 7]
    = 2

    Solve for the remainder of (19^98)/7?

    Rem [19^98/7]
    = Rem [(-2)^98/7]
    = Rem [2^98/7]
    = Rem [ 2^96 * 2^2 / 7]
    = Rem [ 8^32 * 4 / 7]
    = Rem [1 * 4 / 7]
    = 4

    What is the remainder when 7^2015 is divided by 9?

    Rem [7^2015 / 9]
    = Rem [(-2)^2015 / 9]
    = Rem [ 4 * (-2)^2013 / 9]
    = Rem [ 4 * (-8)^671 / 9]
    = Rem [ 4 * 1 / 9]
    = 4

    What is the remainder when 2014^2015 is divided by 9?

    Rem [2014^2015 / 9]
    = Rem [(-2)^2015 / 9]
    = Rem [ 4 * (-2)^2013 / 9]
    = Rem [ 4 * (-8)^671 / 9]
    = Rem [ 4 * 1 / 9]
    = 4

    What is the remainder when 2^2003 is divided by 17?

    Rem [2^2003 / 17]
    = Rem [2^2000 * 8 /17]
    = Rem [16^500 * 8 / 17]
    = Rem [(-1)^500 * 8 / 17]
    = Rem [ 1*8/17]
    = 8

    What is the remainder if 7^121 is divided by 17?

    Rem [7^121 / 17]
    = Rem [7^120 * 7 / 17]
    = Rem [49^60 * 7 / 17]
    = Rem [(-2)^60 * 7 / 17]
    = Rem [ 16^15 * 7 /17]
    = Rem [ (-1)^15 * 7 / 17]
    = Rem [ (-7) / 17]
    = 10

    Type: Simplifying the dividend (Multiple divisors)

    What is the remainder of 2^90/91?

    This looks a little difficult if you do not any theorems for finding out remainders. Let us take a simpler approach and break down the problem into smaller parts.

    These type of questions become really simple if you understand the concept of negative remainders. Always try and reduce the dividend to 1 or -1.

    91 = 7 * 13

    Let us find out Rem[2^90/7] and Rem[2^90/13]
    We will combine them later.

    Rem [2^90/7]
    = Rem [ (2^3)^30 / 7]
    = Rem [ 8^30 / 7]
    = Rem [1^30 / 7]
    = 1

    Rem [2^90/13]
    = Rem [ (2^6)^15 / 13]
    = Rem [ 64^15 / 13]
    = Rem [ (-1)^15 / 13]
    = -1 from 13
    = 12

    So, our answer is a number which leaves a remainder of 1 when divided by 7 and it should leave a remainder of 12 when divided by 13.

    Let us start considering all numbers that leave a remainder of 12 when divided by 13
    => 12 (leaves a remainder of 5 from 7. Invalid)
    => 25 (leaves a remainder of 4 from 7. Invalid)
    => 38 (leaves a remainder of 3 from 7. Invalid)
    => 51 (leaves a remainder of 2 from 7. Invalid)
    => 64 (leaves a remainder of 1 from 7. Valid. This is our answer)

    What is the remainder when 128^1000 is divided by 153?

    153 = 9*17
    128^1000 = 2^7000

    Let us find out Rem[2^7000/9] and Rem[2^7000/17]
    We will combine them later.

    Rem[2^7000/9]
    = Rem [ 2^6999 x 2 / 9]
    = Rem [ 8^2333 x 2 / 9]
    = Rem [ (-1)^2333 x 2 / 9]
    = Rem [ (-1) x 2 / 9]
    = - 2 from 9
    = 7

    Rem[2^7000/17]
    = Rem [16^1750 / 17]
    = Rem [ (-1)^1750 / 17]
    = 1

    So, our answer is a number which leaves a remainder of 7 when divided by 9 and it should leave a remainder of 1 when divided by 17.

    Let us start considering all numbers that leave a remainder of 1 when divided by 17
    => 18 (leaves a remainder of 0 from 9. Invalid)
    => 35 (leaves a remainder of 8 from 9. Invalid)
    => 52 (leaves a remainder of 7 from 9. Valid. This is our answer)

    What is the remainder when 15^40 divided by 1309?

    1309 = 7 * 11 * 17

    Let us find out Rem[15^40/7], Rem[15^40/11] , and Rem[15^40/17]
    We will combine them later.

    Rem[15^40/7]
    = Rem [1^40/7]
    = 1

    Rem[15^40/11]
    = Rem [4^40/11]
    = Rem [256^10/11]
    = Rem [3^10/11]
    = Rem [243^2/11]
    = Rem [1^2/11]
    = 1

    Rem[15^40/17]
    = Rem [(-2)^40/17]
    = Rem [16^10/17]
    = Rem [(-1)^10/17]
    = 1

    So, our answer is a number which leaves a remainder of 1 when divided by 7, 11, and 17
    Such a number is 1 itself and that is our answer.

    Type: Taking values common from Dividend and Divisor

    What is the remainder of the value 39 to the power 198 divided by 12?

    Rem [39^198 / 12]
    = Rem [3^198 / 12]
    = 3 * Rem[3^197 / 4]
    = 3 * Rem[(-1)^197 / 4]
    = 3 * Rem[-1 / 4]
    = 3 * 3
    = 9

    What is the remainder when 3^164 is divided by 162?

    To solve this, you need to know that Rem [ka/kb] = k Rem[a/b]

    We need to find out
    Rem [3^164/162]
    = Rem [(3^4 x 3^160) / (3^4 x 2)]
    = 3^4 Rem [3^160/2]
    = 3^4 Rem [1^160/2]
    = 3^4 x 1
    = 81

    What is the remainder when 21! is divided by 361?

    Rem [21!/361]
    = Rem [(21 * 20 * 19 * 18!)/361]
    Using Rem [ka/kb] = k Rem[a/b]
    = 19 Rem [(21 * 20 * 18!)/19]

    Using Wilson's Theorem says For a prime number 'p' Rem [ (p-1)! / p] = p-1
    = 19 Rem [(21 * 20 * 18)/19]
    = 19 Rem [(2 * 1 * (-1))/19]
    = 19 * (-2)
    = -38
    = 323

    What is the remainder when 2(8!)-21(6!) divides 14(7!) +14(13!)?

    We need to find out Rem [(14(7!) + 14(13!)) / (2(8!) - 21(6!)) ]
    Let us try and simplify the divisor
    2(8!) - 21(6!)
    = 2 * 8 * 7! - 3 * 7 * 6!
    = 16 * 7! - 3 * 7!
    = 13 * 7!

    Rem [(14(7!) + 14(13!)) / 13*7! ]
    = Rem [14(7!) / 13(7!)] + Rem [14(13!) / 13(7!)]
    = 7! * Rem[14/13] + 0
    = 7! * 1
    = 7!


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