What is the sum of digits of highest 4 digit number which when divided

Question

What is the sum of digits of highest 4 digit number which when divided by 6, leaves a remainder 5, when divided by 7, leaves a remainder 6, when divided by 8 leaves a remainder 7 and when divided by 9 leaves a remainder 2

A. 19
B. 20
C. 21
D. 22
E. 23

whats wrong in this :

the number is basically divisible by 6+5 =11, 7+6 = 13, 7+8 = 15, 9+2 =11
so number is = 11*13*15*k
=2145*k such that <5digit
so multiply 4
8,580

hence 21

however the answer is 20!!

Krunaal · Answer

Let the 4-digit number be n

We are given,

n = 6a + 5 ........(i)
n = 7b + 6 .........(ii)
n = 8c + 7 ..........(iii)
n = 9d + 2 ..........(iv)

Looking at the pattern of (i), (ii), and (iii): They all leave remainder 1 less than the divisor,

So the number is 1 less than a multiple of 6, 7, and 8

LCM of 6, 7, and 8 = 168

Therefore,  n = 168k -1  .......... (v)

Now equating (iv) and (v)

168k - 1 = 9d + 2 =>  This tells us we want to find the values of k such that 168k - 1 leaves remainder 2 when divided by 9

168k - 1 ≤ 9999
=> 168k ≤ 10000 
=> k ≤ 59 (since 168*59 = 9912)

Try subs. k = 59 in (v)
n = 9912 - 1 = 9911

Check if we get a remainder of 2 dividing 9911 by 9

9*1101 = 9909 +  remainder 2

Yes! We do get remainder 2, so 9911 is the 4-digit number. Now, sum of its digits = 9+9+1+1 =  20

Answer B.

rak08 · Answer

Whats wrong with this

6+5 = 11, 7+6 = 13, 8+7 = 15, 9+2 =11
so its a multiple of 11*13*15
11*13*15k <10000
2145k<10000
k = 4
ans = 8580 
8+5+8+0 = 21

culpavoluptatum · Answer

This question looks tricky at first and everyone starts solving this with the same method of taking LCM of 6, 7, 8 and 9. But if I say there is 2 second solution for this question will you believe me. The solution is basically hidden in the question.

The are not asking for the largest 4 digits but the sum of the digits. And we also have a divisor whose divisibility rule is defined by the sum of digits, which is the number 9. As the question has asked the largest 4 digit number should satisfy all the 4 rules defined in question for numbers 6,7,8 and 9.
 => And for 9 the rule says that when divided by 9 the remainder should be 2.  So we can look in the answer choice which sum when divided by 9 gives a remainder of 2. And the answer is 20

Option B

What is the sum of digits of highest 4 digit number which when divided

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What is the sum of digits of highest 4 digit number which when divided

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