BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

"Length of integer" (prime number properties)

This topic has expert replies
Post new topic Post Reply
Previous Topic Next Topic
User avatar
Newbie | Next Rank: 10 Posts
Posts: 9
Joined: Thu Sep 15, 2011 9:41 am

"Length of integer" (prime number properties)

by JohnRapp » Sun Nov 13, 2011 9:48 am
For any integer k > 1, the term "length of an integer" refers to the number of positive prime factors, not necessarily distinct, whose product is equal to k. For example, if k = 24, the length of k is equal to 4, since 24 = 2 × 2 × 2 × 3. If x and y are positive integers such that x > 1, y > 1, and x + 3y < 1000, what is the maximum possible sum of the length of x and the length of y?

A. 5
B. 6
C. 15
D. 16
E. 18

I'm hoping this post can provide alternative explanations to the OA explanation.

OA = [spoiler](D) 16[/spoiler]
Top
Source: — Problem Solving |
User avatar
Legendary Member
Posts: 588
Joined: Sun Oct 16, 2011 9:42 am
Location: New Delhi, India
Thanked: 130 times
Followed by:9 members
GMAT Score:720

by rijul007 » Sun Nov 13, 2011 10:14 am
for a number to have maximum length.. it should have the highest no of prime factors not necessarily distinct

this number n should have max nuber of 2s as a prime factor

x>1
y>1

x = 2^a [a is the length of x]
y = 2^b

x+3y<1000
2^a + 3(2)^b <1000

highest value of 2^a that can fit into this is 512

2^a = 512
a = 9

3(2)^b <1000-512
2^b < 488/3
2^b < 162
highest value of 2^b can be 128
b = 7

Maximum sum of length of a and length of b = a + b =9+7 = 16


Option D
Top
User avatar
Legendary Member
Posts: 588
Joined: Sun Oct 16, 2011 9:42 am
Location: New Delhi, India
Thanked: 130 times
Followed by:9 members
GMAT Score:720

by rijul007 » Sun Nov 13, 2011 10:15 am
for a number to have maximum length.. it should have the highest no of prime factors not necessarily distinct

a number should have max nuber of 2s as a prime factor for the max length..

x>1
y>1

x = 2^a [a is the length of x]
y = 2^b

x+3y<1000
2^a + 3(2)^b <1000

highest value of 2^a that can fit into this is 512

2^a = 512
a = 9

3(2)^b <1000-512
2^b < 488/3
2^b < 162
highest value of 2^b can be 128
b = 7

Maximum sum of length of x and length of y = a + b =9+7 = 16


Option D
Top
Post new topic Post Reply
• Page 1 of 1