\(K\)-numbers are positive integers with only \(2\)'s as their digits. For example, \(2, 22,\) and \(222\) are \(K\)

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\(K\)-numbers are positive integers with only \(2\)'s as their digits. For example, \(2, 22,\) and \(222\) are \(K\)-numbers. The \(K\)-weight of a number \(n\) is the minimum number of \(K\)-numbers that must be added together to equal \(n.\) For example, the \(K\)-weight of \(50\) is \(5,\) because \(50 = 22 + 22 + 2 + 2 + 2.\) What is the K-weight of \(600?\)

A. \(10\)
B. \(11\)
C. \(12\)
D. \(13\)
E. \(14\)

The OA is A

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To represent 600 as the sum of K-numbers, we need to find the minimum number of K-numbers that must be added together to equal 600.

The largest K-number less than or equal to 600 is 222. Therefore, we start by dividing 600 by 222:

600÷222=2 remainder 156

This tells us that we can represent 600 as 2 times 222 plus a remainder of 156.

Next, we check how many times we can represent the remainder of 156 using the next largest K-number, which is 22:

156÷22=7 remainder 2

This tells us that we can represent 156 as 7 times 22 plus a remainder of 2.

Finally, the remainder of 2 can be represented using the smallest K-number, which is 2 itself.

So, to represent 600 as the sum of K-numbers, we need:

2 times 222
7 times 22
1 time 2
Therefore, the K-weight of 600 is

2+7+1=10.

The correct answer is A

Bernard Baah
MS '05, Stanford
GMAT and GRE Instructor
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