remainder and min max

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buoyant: Master | Next Rank: 500 Posts; Posts: 106; Joined: Sat Mar 02, 2013 4:29 pm; Thanked: 4 times

by buoyant » Mon Jan 19, 2015 12:30 pm

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

[spoiler]OA:B[/spoiler]

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Source: Beat The GMAT — Problem Solving | Mon Jan 19, 2015 12:30 pm

MartyMurray: Legendary Member; Posts: 2135; Joined: Mon Feb 03, 2014 9:26 amFollowed by: 140 members; Location: https://martymurraycoaching.com/; Thanked: 955 times; GMAT Score:800

by MartyMurray » Mon Jan 19, 2015 1:19 pm

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

[spoiler]OA:B[/spoiler]

Well, x has to be a least 3 for a remainder of 3 when x is divided by y.

So going to say x = 3.

y has to be greater than 3 for x/y to generate a remainder of 3, and y has to be at least 8 for y/z to generate a remainder of 8.

So going to say y = 8.

Z has to be greater than 8 for y/z to generate a remainder of 8. Lowest possible integer greater than 8 is 9.

So going with z = 9.

x + y + z = 3 + 8 + 9 = 20

Choose B.

Marty Murray
Perfect Scoring Tutor With Over a Decade of Experience
MartyMurrayCoaching.com
Contact me at [email protected] for a free consultation.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon Jan 19, 2015 1:24 pm

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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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Mon Jan 19, 2015 1:44 pm

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

When it comes to remainders, we have a nice rule that says:

If N divided by D, leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.

For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

-------------------------

When y is divided by z the remainder is 8
According to the above rule, the possible values of y are: 8, 8+z, 8+2z, 8+3z...
We want to MINIMIZE the value of y, so y = 8

When x is divided by y the remainder is 3
Now that we know that y = 8, we can write: When x is divided by 8 the remainder is 3
According to the above rule, the possible values of x are: 3, 3 + 8, 3 + 2(8)...
We want to MINIMIZE the value of x, so x = 3

What about z????
We already know that, when y is divided by z, the remainder is 8
There's a rule that says, when positive integer N is divided by positive integer D, the remainder R is such that 0 < R < D
So, for the given information, we can conclude that 0 < 8 < z
We want to MINIMIZE the value of z, so z = 9

So, the smallest possible value of x+y+z =3 + 8 + 9
= 20
= B

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

by Matt@VeritasPrep » Sun Jan 25, 2015 4:28 pm

Marty Murray wrote:Well, x has to be a least 3 for a remainder of 3 when x is divided by y.

So going to say x = 3.

Well ... this is mathematical Catch-22, where anyone who could follow it would already know what to do. For the perplexed, what Marty is doing here is exploiting the fact that if we're dividing integers in the remainder system, we have

x/y = 0, with remainder x

if y > x > 0. For instance, 3/5 = 0, with remainder 3; 4/7 = 0, with remainder 4, etc. So x is at least 3 greater than a multiple of y, and since we want the minimum value of x, we can make x = 3 and "a multiple of y" = 0, since 0 is a multiple of any integer.

This is worth delving into a bit, as most students will struggle with it.

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