Quant Boosters - Anubhav Sehgal, NMIMS Mumbai - Set 2
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Solve and Share your method for this one
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Q4) X is a set of all natural numbers with 4 factors such that sum of all its factors excluding number itself is 31. Find the sum of all such numbers
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Case 1 : N = a * b
Factors will be 1,a,b,ab
1 + a + b = 31
a + b = 30
a,b = 7,23 ; 11,19 ; 13,17 => N = 161,209,221Case 2 : N = a^3
Factors : 1,a,a^2,a^3
1 + a + a^2 = 31
a(a + 1) = 30
a = 5 is the only case
N = 125Sum = 716
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Q5) Find the remainder when C(58,29) is divided by 29.
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C(2n,n) = C(n,0)^2 + C(n,1)^2 + .. + C(n,n)^2
For our question,
C(58,29) mod 29
= C(29,0)^2 + C(29,1)^2 + .. + C(29,29) mod 29
All terms except first and last are divisible by 29.
Remainder = C(29,0)^2 + C(29,29)^2
= 1 + 1
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Q6) How many numbers from 1 to 100 are NOT divisible by 2 or 3 or 5 or 7 ?
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105 * 1/2 * 2/3 * 4/5 * 6/7 - {101,103} = 22
Why 105 ?
Step 1 : Find numbers not divisible by 2,3,5 or 7 by simple Euler number's application.
Step 2 : Remove numbers like 103 which being not divisible by 2,3,5 or 7 were removed in euler but do not belong to our range of 1-100 for which answer is to be calculated.
105 Chosen as it is the closest(to100) multiple of LCM of uncancelled denominators 5,7
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Q7) HCF of 2 numbers is 12 and their sum if 144. Find the maximum value of their product
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12(a + b) = 144
a + b = 12
As near to each other as possible with they being coprime to each other.
5, 7
Max product = (12 * 5) * (12 * 7) = 60 * 84 = 5040.
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Q8) Highest power of 8 in 17! + 18! + 19! + . . . + 100!?
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17!(1 + 18 + 18 * 19 + ... 18 * 19 * 20 * ..100)
17!(1 + even)
17! * odd
Find power of 2 and subsequently 8 in 17!
Power of 2 : 8 + 4 + 2 + 1 = 15
Power of 8 : [15/3] = 5
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Q9) What is the highest power of 3 available in the expression 58! - 38!
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38!(58 * 57 * .. * 39 - 1)
Power of 3 in 38! = 12 + 4 + 1 = 17
Power of 3 inside bracket = 0
Highest power to divide 58! - 38! = 17
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Q10) Remainder when 26^57 is divided by 29?
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E(29) = 28
26^57 mod 29
26^56 * 26 mod 29
26
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Q11) How many scalene triangles are there for which the lengths of all sides are integers & the perimeter is 24 cm?
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For this particular case, [(n - 6)^2/48] where [.] nearest integer function.
[(24 - 6)^2 / 48 ] = 7
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Q12) The number of 4-letter words that can be formed out of the letters of the word FIITJEE is
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F,I-2,T,J,E-2
All different : 5c4 * 4! = 120
2 same 2 different : 2c1 * 4c2 * 4!/2! = 144
2 same 2 same : 2c2 * 4!/2!2! = 6
Total : 270
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Q13) Find the sum of the first 125 terms of sequence 1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2 ...
