What is the greatest five-digit positive integer which when divided by

Question

What is the greatest five-digit positive integer which when divided by 12, 15, 21, 25 and 28 leaves 5, 8, 14, 18 and 21 as remainders, respectively ?

(A) 97693
(B) 98693
(C) 98696
(D) 98700
(E) 98796

ThatDudeKnows · Answer

Use the answer choices!

The easiest number to divide by out of the ones listed is 25.

(A) 97693 - 97675 is divisible by 25, so 97693 leaves a remainder of 18. Keep it.
(B) 98693 - 98675 is divisible by 25, so 98693 leaves a remainder of 18. Keep it.
(C) 98696 - 98675 is divisible by 25, so 98696 leaves a remainder of 21. Wrong.
(D) 98700 - remainder 0 when divided by 25. Wrong.
(E) 98796 - 98775 is divisible by 25, so 98796 leaves a remainder of 21. Wrong.

Okay, so we are down to A and B.

Let's just test.

If we divide 97693 by 12, we get a remainder of 1, so A is wrong.

Answer choice B.

SnowWave · Answer

eliminate C,D & E as when dividing by 12 you will not get odd number 5 as remainder.
Test a & b by dividing with 15 . Option B will give Remaidner 8 which is correct.

GMATWhizTeam · Answer

Notice that there is constant difference between each divisor and the respective remainder. 
For example, when divisor is 12, remainder is 5; the difference between 12 and 5 is 7.
Similarly, when divisor is 15, remainder is 8; the difference between 15 and 8 is 7.

In such a case where a given number, say N, is divided by many divisors, say a, b, c. etc., yielding different remainders in such a way that the difference between the divisor and remainder is constant, the number N can be expressed as,

N = LCM (a, b, c…) k – (Common difference), where k is a positive integer.

Therefore, the numbers we are looking for are of the form,

N = LCM (12, 15, 21, 25, 28) k – 7

LCM (12, 15, 21, 25, 28) = 2100.

Therefore, N = 2100 k – 7; in other words, N + 7 = 2100k.

Since we need the greatest five digit number that satisfies the above constraint, we can divide 100,000 by 2100 and subtract the remainder.

When 100,000 is divided by 2100, the remainder is 1300. This means, the greatest five-digit integer which is a multiple of 2100 = 100,000 – 1300 = 98700

Therefore, N + 7 = 98700 and hence, N = 98693.

The correct answer option is B.

BrushMyQuant · Answer

Positive integer which when divided by 25 leaves 18 as remainder

=> Units digit will be 5 + 8 or 0 + 8, i.e. 3 or 8

ELIMINATE C, D, E

Positive integer which when divided by 15 leaves 8 as remainder

=> When the number is divided by 3 then it will leave remainder of remainder of 8 / 3. i.e. 2

Remainder of a number by 3 = Remainder of sum of digits of the number by 3

(A) 97693

Sum of digits = 9 + 7 + 6 + 9 + 3 = 34
Remainder by 3 = Remainder of 34 by 3 = 1
=>   ELIMINATE

(we don't need to check as we have already eliminated all options)

(B) 98693

Sum of digits = 9 + 8 + 6 + 9 + 3 = 35
Remainder by 3 = Remainder of 35 by 3 = 2
 =>   TRUE

So.  Answer will be B 
Hope it helps!

Watch the following video to MASTER Divisibility Rules

https://www.youtube.com/watch?v=D9ZWVUyvFvM

What is the greatest five-digit positive integer which when divided by

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