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A number when divided by 4 and 5 leaves remainders 1 and 4

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A number when divided by 4 and 5 leaves remainders 1 and 4

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Source: Veritas Prep

A number when divided by 4 and 5 leaves remainders 1 and 4 respectively. What will be the remainder the remainder when this number is divided by 20?

A. 0
B. 3
C. 4
D. 9
E. 17

The OA is E.

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BTGmoderatorLU wrote:
Source: Veritas Prep

A number when divided by 4 and 5 leaves remainders 1 and 4 respectively. What will be the remainder the remainder when this number is divided by 20?

A. 0
B. 3
C. 4
D. 9
E. 17

The OA is E.
Say the number is N, when N is divided by 4, the quotient is p and the remainder is 1, and when N is divided by 5, the quotient is q and the remainder is 4.

Thus,

N = 4p + 1 and N = 5q + 4

=> 4p + 1 = 5q + 4

4p = 5q + 3

p = (5q + 3)/4 = 4q/4 + (q + 3)/4 = q + (q + 3)/4

Since p and q are nonnegative integers, (q + 3) must be divisible by 4. One of the possible values of q = 1.

@q = 1, we have N = 5q + 4 = 5*1 + 4 = 9

Thus, the remainder when this number (= 9) is divided by 20 = 9.

The correct answer: D

Hope this helps!

-Jay
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BTGmoderatorLU wrote:
Source: Veritas Prep

A number when divided by 4 and 5 leaves remainders 1 and 4 respectively. What will be the remainder when this number is divided by 20?

A. 0
B. 3
C. 4
D. 9
E. 17
$$\eqalign{
& N = 4Q + 1 \cr
& N = 5K + 4 \cr} $$
$$? = R\,\,\,\left( {0 \le R \le 19} \right)\,\,\,,\,\,{\rm{where}}\,\,\,N = 20J + R\,\,\,\left( {J\,\,{\mathop{\rm int}} } \right)\,$$
$$Q,K,J,R\,\,\,{\rm{ints}}$$

All alternative choices are numerical, hence we are allowed to explore a particular viable case!

$${\rm{Take}}\,\,N = 9\,\,\,\,\left( {Q = 2,\,\,K = 1} \right)\,\,\,\, \to \,\,\,\,\,J = 0\,\,\,{\rm{and}}\,\,\,\,? = R = 9\,\,$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

_________________
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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BTGmoderatorLU wrote:
Source: Veritas Prep

A number when divided by 4 and 5 leaves remainders 1 and 4 respectively. What will be the remainder the remainder when this number is divided by 20?

A. 0
B. 3
C. 4
D. 9
E. 17

The OA is E.
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.

A number when divided successively by 4 leaves a remainder 1
Possible values of the number are: 1, 5, 9, 13, 17, 21,...

A number when divided successively by 5 leaves a remainder 4
Possible values of the number are: 4, 9...STOP!

Both lists contain 9, so this could be the number.

What will be the remainder when this number is divided by 20?
9 divided by 20 = 0 with remainder 9

Answer: D

Cheers,
Brent

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BTGmoderatorLU wrote:
Source: Veritas Prep

A number when divided by 4 and 5 leaves remainders 1 and 4 respectively. What will be the remainder when this number is divided by 20?

A. 0
B. 3
C. 4
D. 9
E. 17
Now let´s avoid street fighting and embrace BJJ, a.k.a real MATH !

$$\eqalign{
& N = 4Q + 1 \cr
& N = 5K + 4 \cr} $$
$$? = R\,\,\,\left( {0 \le R \le 19} \right)\,\,\,,\,\,{\rm{where}}\,\,\,N = 20J + R\,\,\,\left( {J\,\,{\mathop{\rm int}} } \right)\,$$
$$Q,K,J,R\,\,\,{\rm{ints}}$$

$$\left\{ \matrix{
N = 4Q + 1\,\,\,\, \Rightarrow \,\,\,\,5N = 20Q + 5 \hfill \cr
N = 5K + 4\,\,\, \Rightarrow \,\,\,\,4N = 20K + 16 \hfill \cr} \right.\,\,\,\,\mathop \Rightarrow \limits^{\left( - \right)} \,\,\,\,\,\,\,N = 20\left( {Q - K} \right) - 11\,\,\,\,\,\, \Rightarrow \,\,\,\,N - 9 = 20\left( {Q - K} \right) - 20 = 20\left( {Q - K - 1} \right)\,$$

$$N - 9 = 20J\,\,,\,\,\,J = Q - K - 1\,\,{\mathop{\rm int}} \,\,\,\,\, \Rightarrow \,\,\,\,\,N = 20J + 9\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = 9\,\,$$


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

_________________
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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Please fix the prompt, which has been transcribed incorrectly.
It should include the word in red:

Quote:
A number when divided successively by 4 and 5 leaves remainders 1 and 4, respectively. What will be the remainder when this number is divided by 20?

A. 0
B. 3
C. 4
D. 9
E. 17

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Post
BTGmoderatorLU wrote:
Source: Veritas Prep

A number when divided by 4 and 5 leaves remainders 1 and 4 respectively. What will be the remainder the remainder when this number is divided by 20?

A. 0
B. 3
C. 4
D. 9
E. 17

The OA is E.
We need to find a number that when divided by 4 leaves a remainder of 1 and when the quotient from this division is divided by 5 leaves a remainder of 4. Let’s represent this number by n.

Since our number produces a remainder of 1 when divided by 4, it must be true that n = 4p + 1 for some integer p.

Since the quotient from the previous division, which is p, produces a remainder of 4 when divided by 5, we have p = 5q + 4. Let’s substitute this expression of p into the previous equation:

n = 4p + 1

n = 4(5q + 4) + 1

n = 20q + 16 + 1

n = 20q + 17

Finally, since 20q is divisible by 20, the remainder from the division of n by 20 is 17.

Answer: E

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