Quant Boosters - Hemant Malhotra - Set 9
Q26) If a and b are natural numbers with no common prime factor and c is the greatest common divisor of (a+ b) and (a^2 +b^2) then how many values can c take?
e) Cannot be determined
gcd(a,b) = 1.
now c will divide either 2 or ab
Case 1:c divided ab
so c either divides a or b ,If c divides a it must divide a as well
so c will divide a and b both so c=1
Case 2: c divides 2.
c=1 or 2
so value of c could be 1 or 2
Q27) Digital sum of 1! + 2! + ... + 10 !
1!+2!+3! mod 9=0
7!+8!=7!(1+8 ) so mod9=0
9!+10! multiple of 9=0
so digital sum=9
Q28) a + b + c + d = 21, number of solutions such that a, b, c and d are distinct natural numbers.
Method 1 :
so 4x + 3y + 2z + w = 11
(6+5+3+2) + (4+3+1) + (2+1) = 27
So, 24*27 = 648 solutions
This is just simple counting
Method 2 :
so total solution =17+4-1c4-1=20c3
now we want all distinct cases
so remove cases
case1- when three values are equal
so a wiill vary from 1 to 6 so 6 soulution
and total ways to arrange them 4!/3!=4
case2- when two are equal
so 45 cases but this will include cases when three are equal so remove that cases so 39 cases a
and way to arrange them 4!/2!*2!=12
so 20c3 - 4 * 6 - 12 * 39 = 648
Q29) Consider a pair (a,b) of natural numbers satisfying a + b^2 +c^3 = abc where c is the greatest common divisor of a and b .Then , how many such pairs are possible?
let a = Mc, b = Nc, so that M and N are coprime.
Then: Mc + N^2c^2 +c^3 = MNc^3.
M + N^2c + c^2 = MNc^2
So c must divide M. Put M = M'c, then M' + N^2 + c = M'Nc^2.
So M' = (N^2 + c)/(Nc^2 - 1).
So M'c^2 = N + (N+c^3)/(Nc^2 - 1).
So (N + c^3)/(Nc^2 - 1) is an integer.
If c = 1, (N+1)/(N-1) can only be an integer for N = 2 or 3. so (a, b) = (5, 2) and (5, 3).
let c > 1
bcz (N + c^3)/(Nc^2 - 1) is an integer and positive, we must have N + c^3 > = Ng^2 - 1, so N < = (c^3 + 1)/(c^2 - 1). If c = 2, then N < = 3. Then N = 1 gives the solution (a, b) = (4, 2), N = 2 gives (N + c^3)/(Nc^2 - 1) non-integral and hence no solution,
N = 3 gives the solution (a, b) = (4, 6). and for c > 2 there will be no solution so number of values = 4
Q30) The positive integer N has exactly six distinct integer divisors inclusing 1 and N. The product of five of these is 648. Which of the following must be a divisor of N
Method 1 -
Product of factors = N (number of factors/2))
let 6th factor = k
so k * 648 = N^3
so k * 3^4 * 2^3 = N^3
RHS is prefect cube so LHS shuld be perfect cube
so k = 3^2 = 9
Method2 : N = 2 * 3^2
648 = 1 * 2 * 3 * 6 * 18
remaining 9 so OA = 9
Livingston Daimari last edited by
@hemant_malhotra I don't understand how 248 came