Algebra Practice Gym - Anubhav Sehgal, NMIMS Mumbai


  • NMIMS, Mumbai (Marketing)


    How many pairs of non-negative integers exist, such that the difference between their product and sum is 72?

    ab - a - b = 72
    ab - a - b + 1 = 73
    a(b - 1) - 1(b - 1) = 73
    (a - 1)(b - 1) = 73 = 1 * 73 = -1 * -73
    a, b = 2, 74 ; 74, 2 ; 0, -72 ; -72, 0

    Non negative integer pairs = 1 Unordered or 2 ordered
    Total integral solutions = 2 unordered or 4 ordered

    Since it just says pairs of non-negative integers without explicitly declaring them as a,b or x,y so we need to take the unordered solutions since the order in which they occur does not hold relevance for this question. Answer: Only 1 i.e. 2, 74.

    Find the number of positive integers N for which N^2 + 2014 is square of an integer.

    N^2 + 2014 = k^2
    k^2 - N^2 = 2014
    2014 is 4k + 2 form.
    So no solution

    How many right angled triangle can be formed of integral side such that one of the three sides is 84?

    Case 1: When 84 is not the hypotenuse
    x^2 + 84^2 = y^2
    x^2 - y^2 = 84^2 = 2^4 * 3^2 * 7^2
    Number of ways = [f(N/4) – 1]/2
    = [f(2^2 * 3^2 * 7^2) – 1]/2
    = (27 – 1)/2 = 13.

    84 can never be hypotenuse of triangle having integral sides
    as x^2 + y^2 = 84 = 2^2 * 3 * 7 has (4k + 3) primes with odd powers.
    So, 13 ways only.

    How many integers between 1 and 1000 both inclusive can be expressed as difference of squares of two non-negative integers?

    Case 1: Both numbers are even : 2p, 2q
    4p^2 - 4q^2 = 4(p^2 -q^2) = 4k form
    Case 2: Both numbers are odd : 2p + 1 , 2q + 1
    4p^2 + 4p + 1 - 4q^2 - 4q - 1 = 4(p^2 + p - q^2 - q) = 4k form
    Case 3: First even , second odd : 2p, 2q + 1
    4p^2 - 4q^2 - 4q - 1 = 4k - 1 = 4k + 3 form
    Case 4: First odd , second even : 2p + 1, 2q
    4p^2 + 4p + 1 - 4q^2 = 4k + 1 form

    Hence, difference of squares of two non-negative integers can be 4k, 4k+1 or 4k + 3 form but never 4k + 2 form
    So for 1-1000 => (3/4)(1000) = 750 cases it can be expressed as the difference of two squares.

    In how many ways can 7^17 be written as a product of 3 natural numbers?

    a * b * c = 7^17
    a = 7^x , b = 7^y , c = 7^z
    abc = 7^(x + y + z) = 7^17
    x + y + z = 17 => C(17 + 3 - 1,3 - 1) = C(19,2) non negative integral solutions

    But this includes solutions where any two of a, b, c are equal and permutated in 3!/2! = 3 ways while we need ways to write it as product of 3 natural numbers hence should be counting them only once.
    2x + y = 17 => 9 solutions.
    Unordering the solutions: (C(19,2) - 3*9)/3! + 9 = 33 ways; +9 to include them once.

    The number of integer solutions of the equation x^2 + 12 = y^4.

    x^2 + 12 = y^4
    y^4 - x^2 = 12
    (y^2 + x)(y^2 - x) = 12 = 1 * 12 = 2 * 6 = 3 * 4
    No solutions for 1 * 12 and 3 * 4 factor pairs as RHS (=12) is even hence we need just even * even factor pairs.
    (y^2 + x)(y^2 - x) = 2 * 6
    y=+/- 2 and x=+/-2
    4 solutions.

    For how many positive integers x between 1 and 1000, both inclusive, is 4x^6 + x^3 + 5 divisible by 7?

    x^3 mod 7 = 0, 1, -1 only.
    Case 1: (4x^6 + x^3 + 5) mod 7 = 5.
    Case 2: (4x^6 + x^3 + 5) mod 7 = 4(1) + 1 + 5 mod 7 = 10 mod 7 = 3.
    Case 3: (4x^6 + x^3 + 5) mod 7 = 4(1) - 1 + 5 mod 7 = 8 mod 7 = 1.
    Hence zero values.



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