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When x is divided by 10, the quotient is y with a remainder

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When x is divided by 10, the quotient is y with a remainder [#permalink]  12 Aug 2018, 15:45
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Question Stats:

90% (00:43) correct 9% (01:10) wrong based on 42 sessions
When x is divided by 10, the quotient is y with a remainder of 4. If x and y are both positive integers, what is the remainder when x is divided by 5?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
[Reveal] Spoiler: OA

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Retired Moderator
Joined: 07 Jun 2014
Posts: 4803
GRE 1: Q167 V156
WE: Business Development (Energy and Utilities)
Followers: 173

Kudos [?]: 2971 [0], given: 394

Re: When x is divided by 10, the quotient is y with a remainder [#permalink]  17 Aug 2018, 16:03
Expert's post
Explanation

This is a bit of a trick question—any number that yields remainder 4 when divided by 10 will also yield remainder 4 when divided by 5. This is because the remainder 4 is less than both divisors, and all multiples of 10 are also multiples of 5.

For example, 14 yields remainder 4 when divided either by 10 or by 5. This also works for 24, 34, 44, 54, etc.
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Re: When x is divided by 10, the quotient is y with a remainder [#permalink]  07 Jun 2020, 06:46
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sandy wrote:
When x is divided by 10, the quotient is y with a remainder of 4. If x and y are both positive integers, what is the remainder when x is divided by 5?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

$$\frac{x}{10}=y+4$$
x=10y+4
Lets test a couple of numbers (y=1 and y=2)

1) When y=1
x=10*1+4 (14)
$$\frac{14}{5}= 2.4$$ (Remainder is 4)

2) When y=2
x=10*2+4 (24)
$$\frac{24}{5}= 4.4$$ (Remainder is 4)

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Kudos [?]: 4644 [1] , given: 70

Re: When x is divided by 10, the quotient is y with a remainder [#permalink]  07 Jun 2020, 08:05
1
KUDOS
Expert's post
sandy wrote:
When x is divided by 10, the quotient is y with a remainder of 4. If x and y are both positive integers, what is the remainder when x is divided by 5?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

APPROACH #1: Test a possible value of x
When it comes to remainders, we have a nice property 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.

So, from the given information, the possible values of x are: 4, 14, 24, 34, 44, 54,....
If you divide any of these possible x-values by 5, you'll always get a remainder of 4.

APPROACH #2: Use algebra

There's a nice rule that says, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2
Likewise, since 53 divided by 10 equals 5 with remainder 3, then we can write 53 = (10)(5) + 3

From the given information we can write: x = 10y + 4
We can rewrite this as: x = (5)(2x) + 4
We know that (5)(2x) is a multiple of 5, which means (5)(2x) + 4 is 4 MORE THAN a multiple of 5
So, when we divide (5)(2x) + 4 by 5, the remainder will be 4

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Kudos [?]: 47 [1] , given: 4

Re: When x is divided by 10, the quotient is y with a remainder [#permalink]  12 Jun 2020, 17:10
1
KUDOS
Def:

if rem(a/n) = f and rem(b/n) = g, so the remainder of ((a*b)/n) is

f*g, and if f*g is greater than n, the remainder is f*g - n

Now, the problem is:

x = 10*y + 4

If we divide x/5 we have:

remainder(x/f) ) = remainder ((10*y +4)/5)

= rem[(10*y)/5] + remainder (4/5)

= rem [10/5] + 4

= 0 + 4 = 4

Re: When x is divided by 10, the quotient is y with a remainder   [#permalink] 12 Jun 2020, 17:10
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