How many positive integers less than 400 have a remainder of 3 when di

Question

How many positive integers less than 400 have a remainder of 3 when divided by 5 and a remainder of 4 when divided by 6?

(A) 12
(B) 13
(C) 15
(D) 30
(E) 80

GMATinsight · Answer

Let's find first number that satisfies the given criterion

N = 5a + 3 = 6b + 4

i.e. a = (6b+1)/5

b = {4, 9, 14, 19....} and
a = {5, 11, 17, 23....}

i.e. N = {28, 58, 88...}

i.e. values of N are in A.P. with common difference of 30 (LCM of 5 and 6)

Multiples of 30 from 1 to 400 =   = 13
in which first is 30 and last is 390

so N will be {28, 58, 88...388}

Total Values = 13

Answer: Option B

Related Video:
  https://www.youtube.com/watch?v=AOlXE3VRrlc

ivycation2024 · Answer

is there any faster way to solve this?

Krunaal · Answer

n = 5p + 3  => 3, 8, 13, 18, 23, 28, 33, .......

n = 6q + 4  => 4, 10, 16, 22, 28, 34, .......

We can make a common remainder formula for  n  using the above two general formulas

LCM of 5 and 6 => 30

First common remainder for both equations => 28

Therefore,  n = 30k + 28

Now, we want  n < 400

when k is 10, we get 328; when k is 11, we get 358; when k is 12, we get 388

So maximum k can be 12, and minimum 0

Thus  n  will have 12 - 0 + 1 => 13 values

CharuGMATKING · Answer

There are 80 multiples of 5 400 and below.
Therefore there are 79 multiples of 5 excluding 400.
So the numbers will range from 8 as the first one to 398 as the last one.
From these, ones that will give a remainder of 3 would also be 79(add 3 to each multiple).

But we know from the second part that the remainder is even when divided by 6, therefore number has to be even. 
From 8 to 398 there are 39 of them (390/10 as we are skipping the alternate ones which are odd)

From these, for leaving a remainder of 4, we know every 3rd even number will leave a remainder of 4 when divided by 6(with other ones leaving 2 and 0).
Therefore 39/3 = 13 is the final answer.

Not sure if this is faster, but it was super intuitive.

messi16 · Answer

fastest way to solve. Identify the first case where this works which is 28. Then, you know that this occurs every thirty numbers because it is simply the LCM of (5, 6). Thus, count the number of terms. 400-28/30 = 12.4, which is 13 numbers.

How many positive integers less than 400 have a remainder of 3 when di

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How many positive integers less than 400 have a remainder of 3 when di

Prep Toolkit

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My Rewards