buoyant wrote:x, y and z are positive integers 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
When y is divided by z, the remainder is 8.
In other words, y is 8 more than a multiple of z:
y = az + 8, where a is a nonnegative integer.
The smallest possible value of y occurs when a=0:
y = 0*z + 8 = 8.
If y=8, then the statement above becomes:
When y=8 is divided by z, the remainder is 8.
For 8/z to have a remainder of 8, z must be GREATER than 8.
To illustrate:
If z=1, then 8/z = 8/1 = 8 R0.
If z=2, then 8/z = 8/2 = 4 R0.
If z=3, then 8/z = 8/3 = 2 R2.
If z=4, then 8/z = 8/4 = 2 R0.
If z=5, then 8/z = 8/5 = 1 R3.
If z=6, then 8/z = 8/6 = 1 R2.
If z=7, then 8/z = 8/7 = 1 R1.
If z=8, then 8/z = 8/8 = 1 R0.
If z=9, then y/z = 8/9 = 0 R8.
Thus, the smallest possible value of z = 9.
When x is divided by y, the remainder is 3.
In other words, x is 3 more than a multiple of y:
x = by + 3, where b is a nonnegative integer.
The smallest possible value of x occurs when b=0
x = 0*y + 3 = 3.
Thus:
Smallest possible value of x+y+z = 3+8+9 = 20.
The correct answer is
B.
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