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16 Sep 2014, 01:26
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The positive integers \(x\), \(y\), and \(z\) are such that when \(x\) is divided by \(y\), the remainder is 3, and when \(y\) is divided by \(z\), the remainder is 8. What is the smallest possible value of \(x+y+z\)?
A. 12
B. 20
C. 24
D. 29
E. 33
A. 12
B. 20
C. 24
D. 29
E. 33
16 Sep 2014, 01:26
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Official Solution:
The positive integers \(x\), \(y\), and \(z\) are such that when \(x\) is divided by \(y\), the remainder is 3, and when \(y\) is divided by \(z\), the remainder is 8. What is the smallest possible value of \(x+y+z\)?
A. 12
B. 20
C. 24
D. 29
E. 33
The statement "when \(x\) is divided by \(y\), the remainder is 3" can be expressed as \(x = qy + 3\), where \(q\) is the quotient, an integer that is greater than or equal to 0. This equation implies that the smallest value for \(x\) occurs when \(q\) equals 0, giving us \(x = 3\). Essentially, this shows that the smallest possible value of \(x\) will be less than \(y\). For instance, if we divide 3 by 4, we indeed get a remainder of 3.
Similarly, the condition "when \(y\) is divided by \(z\), the remainder is 8" implies that the smallest possible value for \(y\) is 8, therefore \(y < z\). As a result, the smallest possible value of \(z\) must be one more than 8, or 9. For instance, if we divide 8 by 9, we indeed get a remainder of 8.
Therefore, the smallest possible value of \(x+y+z\) is \(3+8+9=20\).
Answer: B
The positive integers \(x\), \(y\), and \(z\) are such that when \(x\) is divided by \(y\), the remainder is 3, and when \(y\) is divided by \(z\), the remainder is 8. What is the smallest possible value of \(x+y+z\)?
A. 12
B. 20
C. 24
D. 29
E. 33
The statement "when \(x\) is divided by \(y\), the remainder is 3" can be expressed as \(x = qy + 3\), where \(q\) is the quotient, an integer that is greater than or equal to 0. This equation implies that the smallest value for \(x\) occurs when \(q\) equals 0, giving us \(x = 3\). Essentially, this shows that the smallest possible value of \(x\) will be less than \(y\). For instance, if we divide 3 by 4, we indeed get a remainder of 3.
Similarly, the condition "when \(y\) is divided by \(z\), the remainder is 8" implies that the smallest possible value for \(y\) is 8, therefore \(y < z\). As a result, the smallest possible value of \(z\) must be one more than 8, or 9. For instance, if we divide 8 by 9, we indeed get a remainder of 8.
Therefore, the smallest possible value of \(x+y+z\) is \(3+8+9=20\).
Answer: B
16 Sep 2016, 02:43
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Karanagrawal
Remember this rule:
If N divided by D, leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
