What is the smallest number that leaves a remainder of 2 when divided

Question

What is the smallest number that leaves a remainder of 2 when divided by 3, 3 when divided by 4, 4 when divided by 5, 5 when divided by 6, and 6 when divided by 7?

A. 419
B. 426
C. 431
D. 433
E. 435

ScottTargetTestPrep · Accepted Answer

Solution:
 
As we can see, all the remainders are 1 less than their respective divisors; therefore, the smallest number is 1 less than the least common multiple (LCM) of all the divisors. Since the LCM of 3, 4, 5, 6, and 7 is 3 x 4 x 5 x 7 = 420, the smallest number that has the required properties is 420 - 1 = 419.

Answer: A

jhavyom · Answer

What is the smallest number that leaves a remainder of 2 when divided by 3, 3 when divided by 4, 4 when divided by 5, 5 when divided by 6, and 6 when divided by 7?

A. 419
B. 426
C. 431
D. 433
E. 435

Let the number be n
So, n+1 is a number, which is divisible by 3, 4, 5, 6, and 7
n+1 = LCM of 3, 4, 5, 6, and 7
n+1 = 420
n = 419

Choice A is the answer.

Ayushi1597 · Answer

Hi,

Could you please explain how we got to the conclusion n+1 is divisible by all the numbers (3,4,5,6,7) ?

jhavyom · Answer

For a number (n) to be divisible by a particular divisor (d), the remainder can be any digit from 0 to d-1

Let us take a divisor.
For example, d = 3

If an integer is divided by d = 3, the remainder can be any digit from 0 to 2(i.e. d-1)

For n = 7, and d = 3, the remainder is 1
For n = 8, and d = 3, the remainder is 2
For n = 9, and d = 3, the remainder is 0 not 3
Note that, you get a number (i.e. 9) that is divisible by 3

So, for a divisor, d = 3, the remainder can be a maximum of 2 and if you add 1 more to the dividend, you will get a remainder of 0
 Moreover, this new dividend will be divisible by 3

Similarly, for divisor, d = 4, the remainder can be any digit from 0 to 3

For n = 10, and d = 4, the remainder is 2
For n = 11, and d = 4, the remainder is 3
For n = 12, and d = 4, the remainder is 0 not 4

So, for a divisor, d = 4, the remainder can be a maximum of 3 and if you add 1 more to the dividend, you will get a remainder of 0
 Moreover, this new dividend will be divisible by 4

So, for a number (n) that leaves a remainder of 2 when divided by 3, 3 when divided by 4, 4 when divided by 5, 5 when divided by 6, and 6 when divided by 7.
if you add 1 to that number, it will leave a remainder of 0 when divided by 3, 0 when divided by 4, 0 when divided by 5, 0 when divided by 6, and 0 when divided by 7.

Moreover, the new number obtained by adding 1 to n (i.e. n+1) will be divisible by 3, 4, 5, 6, and 7 since every digit 3, 4, 5, 6, and 7 leaves a remainder of 0 on dividing n+1.

bv8562 · Answer

The only option which yields remainder 4 when divided by 5 is option A i.e  419 (A)

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

What is the smallest number that leaves a remainder of 2 when divided

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

What is the smallest number that leaves a remainder of 2 when divided

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards