What is the greatest positive four-digit integer which must be added

Question

What is the greatest positive four-digit integer which must be added to 5231 so that the final number becomes divisible by 12, 15, 27, 32 and 40 ?

(A) 9729
(B) 7929
(C) 7829
(D) 7729
(E) 6729

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Archit3110 · Answer

5321+ 4 digit number which should be divisible by 12,15,27,32 , 40
take LCM we get 2^3*3^3*5 ; 4320
5231+7729 ; 12960 is divisible by 4320
option D

zhanbo · Answer

The smallest number that is divisible by 12, 15, 27, 32 and 40 is their LCM (Least Common Multiple), which turns out to be
4320.

4320 is less than 5231.
So, let's start with 4320 * 2 = 8640
8640 - 5231 = 3409, which is not even in the answer choices because the question asks for the greatest positive four-digit integer. 
So 3409 + 4320 = 7729
(If we add another 4320 to 7729, we will get a five-digit integer.)

So the answer is  (D)  7729

Nidzo · Answer

Prime factorising the divisors:

12:    3*4 
 15:   3*5 
 27:    3^3 
 32:   2^5 
 40:   5*2^3

LCM:  2^5 * 3^3 * 5

32 * 27 * 5 = 4320  The answer will be the choice which when added to  5231  yields a multiple of  4320 .

Looking at the sum of  5231  and each of the answer choices, their respective sums will range somewhere between  11 000   and  16 000 . Multiples of  4320  which are 5 digits big will  4320*3  and  4320*4 . However,  4320*4  will take us our of the range of between  11 000   and  16 000 .

4320*3 = 12960  will be the answer.

Glancing at the answer choices: Adding  5231  to (A), (B) & (C) will send us over  12960 . Adding  5231  to (E) will put us under. (D) is the safest place to start.

(D)  5231+7729 = 12 960

Answer D

IanStewart · Answer

We could also notice that, using any answer choice, we'll get a multiple of 5 and 4, because the choices all have the same last two digits. So we probably don't need to worry about 2s and 5s here. But we need also to get a multiple of 3 when we add the right answer choice, and that's not going to happen with most of the choices. The number 5321 gives a remainder of 2 when you divide by 3 (since the sum of its digits is 11), so we need to add something with a remainder of 1 when divided by 3 if we want the sum to be divisible by 3. Summing the digits of each answer choice, only D, 7729, gives a remainder of 1 when divided by 3, so it must be right.

bumpbot · Answer

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What is the greatest positive four-digit integer which must be added

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What is the greatest positive four-digit integer which must be added

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