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22 Jan 2011, 12:31
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D
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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
A. 5
B. 6
C. 15
D. 16
E. 18
22 Jan 2011, 12:49
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ajit257
The length of an integer is essentially the sum of the powers of its prime factors. Take the number 24, for example. 24 = 2^3*3^1, therefore, its length is 4 because 3 + 1 equals 4.
Given the inequality x + 3y < 1,000, to maximize the length of x or y (or in other words, to maximize the sum of the powers of their prime factors), we should minimize their prime bases. The smallest prime base is 2. If x = 2^9 = 512, then its length becomes 9. Therefore, with 512 + 3y < 1,000, we have y < 162.7. The maximum length of y would be 7, since 2^7 = 128.
Thus, the maximum possible sum of the length of x and the length of y is 9 + 7 = 16.
Answer: D.
09 Jun 2012, 22:30
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here is my approach :
we know that : x > 1, y > 1, and x + 3y < 1000,
and it is given that length means no of factors.
for any value of x and y, the max no of factors can be obtained only if factor is smallest no & all factors are equal.
hence, lets start with smallest no 2.
2^1 =2
2^2 =4
2^3=8
2^4=16
2^5=32
2^6=64
2^7=128
2^8=256
2^9=512
2^10 =1024 (opps//it exceeds 1000, so, x can't be 2^10)
so, max value that X can take is 2^9 , for which has "length of integer" is 9.
now, since x =512 , & x+3y<1000
so, 3y<488
==> y<162
so, y can take any value which is less than 162. and to get the maximum no of factors of smallest integer, we can say y=2^7
for 2^7 has length of integer is 7.
SO, combined together : 9+7 = 16.
And is D.
Hope it will help.
we know that : x > 1, y > 1, and x + 3y < 1000,
and it is given that length means no of factors.
for any value of x and y, the max no of factors can be obtained only if factor is smallest no & all factors are equal.
hence, lets start with smallest no 2.
2^1 =2
2^2 =4
2^3=8
2^4=16
2^5=32
2^6=64
2^7=128
2^8=256
2^9=512
2^10 =1024 (opps//it exceeds 1000, so, x can't be 2^10)
so, max value that X can take is 2^9 , for which has "length of integer" is 9.
now, since x =512 , & x+3y<1000
so, 3y<488
==> y<162
so, y can take any value which is less than 162. and to get the maximum no of factors of smallest integer, we can say y=2^7
for 2^7 has length of integer is 7.
SO, combined together : 9+7 = 16.
And is D.
Hope it will help.
