Quant Boosters  Hemant Malhotra  Set 9

For 4 consecutive natural number there will be atleast 2 even numbers so product will be even
= > x is even and x+1 is odd
so 1 is true and 2nd is false
now to decide where n is prime or not
try for first four natural numbers which give a product of 24 and therefore n =25, therefore n is not prime but could be prefect squareOR
n= (a1) * (a) * (a+1)(a+2)
n= (a^2+a1)(a^2+a1)
so n is perfect squareOR
x = 2 * 3 * 4 * 5.
then 120 + 1 = 121 = 11^2so option 3.

Q11) If 2n + 1 and 3n + 1 ( n # 0 ) are perfect squares, find remainder when n is divide by 40 ?

n = 40 so 2n+1 = 81 = 9^2
3n+1 = 121 = 11^2
40 mod 40 = 0

Q12) The sum of 6 natural numbers (not necessarily distinct) is 23. Let M denote the positive difference of the maximum and minimum of the LCM of these 6 numbers. Then total number of divisors that divide M^2 but doesn't divide M are
(a) 22
(b) 33
(c) 18
(d) 26
(e) none

Minimum occurs when the numbers are 6, 6, 6, 2, 2, 1.
Maximum occurs when the numbers are 7, 5, 4, 3, 2, 2.
To achieve min LCM look for max number of hcf among the 6C2 pairs.
So we need number in form of coprime.Thus, M = 414 = 2 * 3^2 * 23, M has 2 * 3 * 2 = 12 divisors
M^2 has 3 * 5 * 3 = 45 divisors.so 45  12 = 33 divisors

Q13) N = 98765432109876543210.... ( 1000 digits ).What is the least positive value of n such that N + n is divisible by 11 ?

Divisiblity rule of 11: Numerals whose alternating sum of digits is divisible by 11 represent numbers divisible by 11. Here the "alternating sum" means we alternate the signs from positive to negative to positive to negative, and so on. 10 to an odd power plus 1 is divisible by 11, and 10 to an even power minus 1 is divisible by 11. This test can also be applied recursively
Example: 12342
1  2 + 3  4 + 2 = 0 so div by 11
Another one, 2131415
2  1 + 3  1 + 4  1 + 5 = 11 so div by 11In given case, make a group of 10 numbers and add all such group, there will be 100 groups so total sum is 9876543210 * 100 = 987654321000
Now use alternate sum approach so remainder is 5 ; so u need to add 5

Q14) Exactly one of the ﬁve numbers listed below is a prime number. Which one is the prime number ?
(a) 999,991
(b) 999,973
(c) 999,983
(d) 1,000,001
(e) 7,999,973

a) (10^3)^2  3^2 = > (1003)(997)
b) (10^2)^3  3^3 = > divisible by (10^2 3) = 97
d) (10^2)^3 + 1^3 = > divisible by 101
e) (2*10^2)^3  3^3 = > divisible by 200 3 = 197so c is prime

Q15) Find the smallest prime number N such that the following is true:
The largest prime factor of N − 1 is A
The largest prime factor of A − 1 is B
The largest prime factor of B − 1 is 7a) 187
b) 347
c) 119
d) 311

use options.
for N = 347
346 = 2 * 173
172 = 4 * 43
42= 2 * 3 * 7

Q16) What are the number of coprimes of y less than y, where y is the largest number with which when 486, 686 and x are divided the remainders are the same and x is the largest 3 digit number which when divided by 3 or 8 leaves a remainder of 2 in each case.
a) 10
b) 20
c) 30
d) 40
e) None of these

x = 24k + 2 = 986
486, 686 and 986 gives a remainder of 86 when divided by 100
y = 100
coprimes = 100 * (1/2)(4/5) = 40

Q17) Let x and y be positive integers such that (x − y) and (x + y) are prime. Then which of the following statements is/are true?
I. x^2 + y^2 cannot be prime.
II. x^2 + y^2 cannot be even.
III. x^2 − y^2 may be even or odd.a) I only
b) II only
c) III only
d) I and III only
e) I and II only

If x and y are integers, then (x − y) and (x + y) are either both even, or both odd.
Since (x − y) and (x + y) are prime numbers, therefore, both are odd, and so one of x and y needs to be an even number and the other has to be an odd number.
(x^2 + y^2) and (x^2 − y^2) will always be odd.
II is true and III is false.
Now consider x = 5 and y = 2
x^2 + y^2 = 25 + 4 = 29 which is a prime number.
I is false.
so option 2

Q18) Find x if x^2 = 444447555561
a) 666661
b) 666669
c) 233331
d) 2111119

digital sum of x^2 = 9
So digital sum of x shud be 3
Option b's digital sum is 3 hence b.

Q19) Let x, y, z be prime numbers in the arithmetic progression such that x > y > z. Which among the following is always true ?
(a) x – z is divisible by 12
(b) x + y is divisible by 8
(c) xz  1 is divisible by 7
(d) atleast two of the above
(e) none of the above

x = 7, y = 5 and z = 3
Option E

Q20) How many ordered and unordered triplets of nonnegative integers (a , b , c) are there such that root(a) + root(b) + root(c) = root(625)