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
[spoiler[OA=E[/spoiler]
Source: Veritas Prep
A number when divided successively by 4 and 5 leaves
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The question means something different from what it actually says, but I can guess what it's trying to say.
When we take our number, "n", and divide it by 4, we get a remainder of 1. So n is exactly 1 greater than some multiple of 4, and we have:
n = 4q + 1
It's here where the wording is confused. We are not now "successively" dividing our number n by 5, so we are not dividing n/4 by 5 (which wouldn't make sense in a remainders question, because n/4 is not an integer at all). Instead we're meant here only to be dividing the resulting quotient by 5, ignoring the remainder. So it's only q in the equation above that we're dividing by 5. Since we get a remainder of 4 when we divide q by 5, and we have:
q = 5k + 4
And now we can just plug that in for q in our first equation above:
n = 4q + 1 = 4(5k + 4) + 1 = 20k + 17
and we can see that n is 17 larger than some multiple of 20, and thus when we divide it by 20, the remainder will be 17.
When we take our number, "n", and divide it by 4, we get a remainder of 1. So n is exactly 1 greater than some multiple of 4, and we have:
n = 4q + 1
It's here where the wording is confused. We are not now "successively" dividing our number n by 5, so we are not dividing n/4 by 5 (which wouldn't make sense in a remainders question, because n/4 is not an integer at all). Instead we're meant here only to be dividing the resulting quotient by 5, ignoring the remainder. So it's only q in the equation above that we're dividing by 5. Since we get a remainder of 4 when we divide q by 5, and we have:
q = 5k + 4
And now we can just plug that in for q in our first equation above:
n = 4q + 1 = 4(5k + 4) + 1 = 20k + 17
and we can see that n is 17 larger than some multiple of 20, and thus when we divide it by 20, the remainder will be 17.
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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, a remainder of 4 remains. Let's represent this number by n.M7MBA wrote: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
[spoiler[OA=E[/spoiler]
Source: Veritas Prep
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 for 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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