3 Questions

[MBA.Aspirant]

1)

If k is an integer, and (35^2-1)/k is an integer, then k could be each of the following, EXCEPT

(A)8
(B) 9
(C) 12
(D) 16
(E) 17


2)

k is a positive integer and 225 and 216 are both divisors of k. If K=2^a x 3^b x 5^c where a, b and c are positive integers, what is the least possible value of a+ b+ c?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 8


3)

If n is an integer and 5^n > 4,000,000 , what is the least possible value of n ?

(A) 7
(B) 8
(C) 9
(D) 10
(E) 1154,000,000

[Frankenstein]

Hi,
1) (35^2-1)/k = 34.36/k Except 16 all other options are factors of 34.36
Hence, D

2)225= 3^2.5^2
216= 2^3.3^3
So, least value of k=2^3.3^3.5^2
So, least value of a+b+c = 3+3+2 = 8
Hence, E

3)5^n > 4.10^6 = 2^2.(2.5)^6
So, 5^(n-6) > 2^8 = 256.
So, n-6>=4 i.e. n>=10

Hence, D

[iamsaurav]

d,e,d

[cans]

If k is an integer, and (35^2-1)/k is an integer, then k could be each of the following, EXCEPT

(A)8
(B) 9
(C) 12
(D) 16
(E) 17

(35^2-1)/k = (35-1)(35+1)/k = 34*36/k
k=8; integer
k=9; integer
k=12; integer
k=16; not integer
k=17; integer
IMO D

[cans]

k is a positive integer and 225 and 216 are both divisors of k. If K=2^a x 3^b x 5^c where a, b and c are positive integers, what is the least possible value of a+ b+ c?

(A) 4
(B) 5
(C) 6
(D) 7
(E) 8

225,216 divisors of k. lcm of 225,216 = 2^3*3^3*5^2
thus k is minimum 2^3*3^3*5^2
thus min a+b+c = 3+3+2=8
IMO E

[cans]

If n is an integer and 5^n > 4,000,000 , what is the least possible value of n ?

(A) 7
(B) 8
(C) 9
(D) 10
(E) 1154,000,000

5^n>2^8 * 5^6
5^n >256*5^6 (5^3=125, 5^4=625 ; as 5^n is greater, we choose 5^4 thus 4+6=01)
n=10
IMO D