What is the sum of all the odd numbers of four digits that are divisib

Question

What is the sum of all the odd numbers of four digits that are divisible by 9?

(A) 2448729

(B) 2478429

(C) 2754000

(D) 2784491

(E) 2796600

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IanStewart · Answer

The smallest four-digit multiple of 9 is 1008 (the smallest four-digit number whose digits sum to 9), but that's not odd, so the smallest number we'll actually be including in our sum is 1017. The largest four-digit odd multiple of 9 is 9999. Odd multiples of 9 are separated by 18, so our sum is

1017 + 1035 + 1053 + ... + 9981 + 9999

which is an equally spaced sum. In an equally spaced list, the average term is always equal to the average of the smallest and largest, so the average term here is (1017+9999)/2 = 11,016/2 = 5508, and the number of terms is (range/space) + 1, so here is (9999-1017)/18 + 1 = 8982/18 + 1 = 500. So the sum is (5508) * (500) = 2754*1000 = 2,754,000.

This is a lot more complicated, computationally, than any similar GMAT question. I'd often check answer choices here -- since we're adding multiples of 9, the answer must be a multiple of 9, so by summing the digits of each answer choice we can quickly rule out impossible choices -- but that only eliminates two options here, so some kind of longer calculation is required.

Gokha · Answer

IanStewart :
 I used a different approach in the end.. but seeing your reply.. there is a quickest way may be less than 1 minute.

We know that any number to be divisible by 9 it's digit sum should be multiple of 9.

Seeing option: (A) 2448729 - sum of digit 36

(B) 2478429 - sum of digit 36

(C) 2754000 - sum of digit 18

(D) 2784491- sum of digit -35 - eliminated

(E) 2796600 - sum of digit - 30 - eliminated.

now we know that there are 500 digits ( Odd multiples of 9 are separated by 18) and our series is 1017 + 1035 + 1053 + ... + 9981 + 9999

this means Odd + Odd ..... till 500 terms which will yield an even number, seeing option, only C fits in the category.

Hence C Answer

IanStewart · Answer

Yes, that's correct -- and if you can see quickly there are 500 numbers in our sum, and you know, for an equally spaced set that sum = n*avg, and avg = (smallest+largest)/2, then you know

sum = 500*(smallest + largest)/2
sum = 250*(some integer)

and you can be sure the answer will be a multiple of 250, and in this question, only one answer choice is. So with these choices, that might be the fastest approach, because we don't even need to check which answers are divisible by 9.

BrushMyQuant · Answer

What is the sum of all the odd numbers of four digits that are divisible by 9?

First number  = 1017 (As 1017 = 999 + 9 + 9)
 Last Number  = 9999
 Common difference, d  = 9*2 = 18 (Alternate multiples of 9)
 Number of terms, n  = (Last term - First Term)/d + 1 =  \frac{9999 - 1017}{18}  + 1 = 499 + 1 = 500
 Sum  = n * (First Term + Last Term)/2 = 500 *  \frac{(1017 + 9999)}{2}  = 2,754,000

So,  Answer will be C 
Hope it helps!

Watch the following video to MASTER Sequence problems

https://www.youtube.com/watch?v=EfNT2JoE5CM

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What is the sum of all the odd numbers of four digits that are divisib

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What is the sum of all the odd numbers of four digits that are divisib

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