A number when divided successively by 4 and 5 leaves remainders 1 and 4 respectively. What will be the remainder when th

BTGModeratorVI wrote: ↑

Wed Dec 16, 2020 12:35 pm

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
Answer: E
Source: Manhattan prep

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

BTGModeratorVI wrote: ↑

Wed Dec 16, 2020 12:35 pm

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
Answer: E

Solution:

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.

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