Quant Boosters - Sagar Gupta - Set 1


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    Euler[224] =96
    99 mod 96 = 3
    57^3 + 55^3 mod 224
    [57+55 ] ( 57^2 + 55^2 - 57*55 ) mod 224
    112 * 3 mod 224 = 112

    Or Use binomial
    (56 + 1)^99 + (56 - 1)^99
    Remainder will be 56 * 99 * 2 or 112(2 * 49 + 1) or 112


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    Q24) Find the remainder when 987698769876.... 400 digits is divided by 31


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    Take [9876] as one digit, so total 100 digits
    Euler(31) = 30, so just take 100mod30=10, ie last ten digits
    Which is 9876 (1 + 10^4 + 10^8 + .... + 10^36) mod 31
    (GP sum inside bracket) mod 31 = 13
    9876mod 31=18
    So 18 * 13 mod 31=17


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    Q25) Sum of 5 terms in an increasing GP is 1031. All the terms are integers. Find the sum of first and the last term of this GP.


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    a + ar + ar^2 + ar^3 + ar^4 = 1031
    a [ 1 + r + r^2 + r^3 + r^4 ] = 1031
    a [ r^5 -1 ] = 1031 [ r -1 ]
    rearrange RHS to express 1031 in the form of r^5 -1
    Look for an integer whose fifth power is more than 1031
    3a [ r^5 - 1 ] = 3093 [ r-1 ]
    5^5 = 3125
    3a [ r^5 - 1 ] = [ 3125 - 32 ] [ r-1 ]
    3a [ r^5 -1 ] = 32 [3125/32 -1 ] [ r-1 ]
    r = 5/2 = 2.5
    3a = 32 ( r-1 )
    3a = 32 (1.5)
    a=16


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    Q26) The no of positive integers n such that (n^6 + 206) is divisible by (n^2 + 2) is ?


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    n^6 + 206 = k ( n^2 + 2 )
    Rearrange :
    n^6 + 8 = k (n^2 + 2) - 198
    (n^2 + 2 ) ( n^4 + 4 - 2n^2 ) = k (n^2 + 2 ) - 198
    Divide throughout by n^2 + 2
    n^4 + 4 - 2n^2 = k - [ 198 / n^2 + 2 ]
    198 / n^2 + 2 must be an integer
    n=1 , n = 2 , n=3 , n=4 , n=8 , n=14 -> Hence 6 Values


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    Q27) X takes 9 and Y takes D days to complete a job working alone. They work on alternate days. If they take exactly the same time irrespective of who starts, how many positive integer values are possible for D ?


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    2m + 18m/d = 18
    m + 9m/d = 9
    d = 9m / 9-m
    m=6 : 54/3 =18
    m=8 : 72 / 1 = 72


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    Q28) Find the sum of all values of x , that satisfy (4x^2 + 15x + 17)/(x^2 + 4x+12) = (5x^2 + 16x+18)/(2x^2 + 5x + 13)


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    m/n = m+p/n+p
    mn + mp = mn + np
    m = n
    4x^2 + 15x + 17 = x^2 + 4x + 12
    3x^2 + 11x + 5 = 0
    Sum of roots = -11/3


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    Q29) The values of the numbers 2^2004 and 5^2004 are written one after another. How many digits are there in all?
    (1) 4008
    (2) 2003
    (3) 2004
    (4) None of these


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    No. of digits = logN + 1 ... here it's 2004(log10) + 1=2005

    Or by pattern : 2^1 5^1 = 25 : 2 digits
    2^2 5^2 = 425 : 3 digits
    2^3 5^3 = 8125 : 4 digits
    2^4 5^4 = 16625 : 5 digits
    2^5 5^5 = 323125 : 6 digits
    ..
    2^2004 5^2004 : 2005 digits


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    Q30) How many integer solutions exist for the equation 5x - 3y = 140 such that x and y are of opposite signs?
    a) 8
    b) 9
    c) 10
    d) 11


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    5x - 3y = 140
    x=25 , x=22 , x=19 , x=16 , x=13 , x=10 , x=7 , x=4 , x=1
    9 solutions


 

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