Quant Boosters  Maneesh  Set 1

Number of factors = 4 * 5 * 10 = 200
Product of factors = 2^300 * 3^400 * 5^900
Number of factors which are divisible by 12 = 2 * 4 * 10 = 80
Product of factors which are divisible by 12 = 2^40 * 3^120 * 5^360 * 12^80
Required product = 2^300 * 3^400 *5^900 / 2^40 * 3^120 * 5^360 * 2^160 *3^80
= 2^100 * 3^200 * 5^540

Q2) N = 2^15 * 3^12 How many factors of N^2 are less than N but do not divide N completely?

Factors of N = 16 * 13 = 208
Factors of N less than N = 208  1 = 207
Factors of N^2 = 31 * 25 = 775
Factors of N^2 less than N = (775  1)/2 = 387
So ans = 387 – 207 = 180

Q3) The numbers 2604,1020 and 4812 when divided by a number N give the same remainder of 12. Find the highest such number N

Since all the numbers are giving remainder 12, (260412),(102012),(481212) are multiples of N
2592, 1008, 4800
So N will be HCF of 2592,1008 and 4800
2592 = 2^5 * 3^4
1008 = 2^4 * 7 * 3^2
4800 = 2^6 * 5^2 * 3
HCF = 2^4 * 3 = 48

Q4) The numbers 400,536 and 645 when divided by a number N gave remainders of 22,23 and 24 respectively.Find the greatest such number N

N will be the HCF of (40022),(53623) and (64524)
HCF(378,513,621) = 27

Q5) The HCF of two numbers is 12 and their sum is 288. How many pairs of such numbers are possible?

Let numbers be 12a, 12b
12(a+b) = 288
a+b = 24 and a and b are co prime
Hence(1,23)(5,19)(7,17)(11,13)
Hence 4 pairs available.

Q6) HCF of 2 numbers are 12 and their product is 31104. How many such numbers are possible?

Let numbers are 12a n 12b
144ab = 31104
ab = 216
a and b are coprime
(1,216)(8,2)
Only two pairs possible.

Q7) Find the Highest five digit number that is divisible by each of the numbers 24, 36, 45 and 60?

LCM (24,36,45,60) => 24 = 2^3 * 3
36 = 3^2 * 2^2
45 = 3^2 * 5
60 = 2^2 * 3 * 5
LCM = 2^3 * 3^2 * 5 = 360
So we have to find highest five digit multiple of 360
99999 mod 360 = 279
so answer 99999279 = 99720

Q8) Find the lowest number which gives a remainder of 5 when divided by any of the numbers 6, 7, 8 ?

LCM(6, 7, 8) = 168
number = 168 + 5 = 173

Q9) What is the smallest number which when divided by 9,18 and 24 which leaves a remainder of 5, 14 and 20 respectively

Q10) In a book store “OM INNOVATIVE BOOK STORE“ is flashed using neon lights. The words are individually flashed at intervals of 8/3, 16/3 , 14/3 and 15/2 seconds respectively and each word is put off after a second. The least time after which the full name of the book store can be read for a second?

Q11) A number when divided by 3, 4, 5 and 6 always leave a remainder 2, but leaves no remainder when divided by 7. What is the lowest such number possible?

LCM(3, 4, 5, 6) = 60
Number of the form 60a + 2
60a + 2 = 7b
lowest such number > 182 when a = 3 & b = 26

Q12) For how many pairs (a,b) of natural numbers is the LCM of a and b 2^3 * 5^7 * 11^13?