The digital sum of a number is the sum of its digits. For how many of

Question

The digital sum of a number is the sum of its digits. For how many of the positive integers 24-125 inclusive is the digital sum a multiple of 7?

A 7 
B 8
C 14
D 16
E 20

mvictor · Answer

is there other way than just listing?

25
34
43
52
59
61
68
70
77
86
95
106
115
124

14 ways..

KarishmaB · Answer

The sequence of 10 numbers from say 30 - 39 or 40 - 49 etc will have an increasing value of the digital sum. These are 10 numbers so depending on the sum with which the sequence starts, there will be 1 or 2 numbers only with a sum which is a multiple of 7 in each range of 10 numbers. If the number in which the digits sum is a multiple of 7 comes within the first 3 numbers, then there will be another such number. Else there won't be.
24 - 29 -> the only number will be 25
30 - 39 -> 1 number only
40 - 49 -> 1 number only
50 - 59, 60 - 69, 70 - 79 -> 2 numbers each so total 6 numbers
80 - 89 -> 1 number only
90 - 99 -> 1 number only
100 - 109 -> 1 number only
110 - 119 -> 1 number only
120 - 125 -> 1 number

Adding all these up, you get 14.

Answer (c)

broall · Answer

No way but listing all numbers, but in better way.

First, we count the 2-digit numbers.
The largest 2-digit numbers is 99, and its digital sum is 18 > 14. Hence, we need to consider 2 cases:

Case 1: The digital sum is 7:

From 24, the next number which has digital sum 7 is 25. Now, we have the list:
25, 34, 43, ..., 61, 70 (there are 7 - 2 + 1 = 6 numbers)

Case 2: The digital sum is 14.

Not that the largest unit digit is 9, so the smallest ten digit is 5. Hence, we have the list:
59, 68, 77, ..., 95 (there are 9 - 5 + 1 = 5 numbers)

Second, we count the 3-digit numbers.
The largest possible number is 125, so the number that has largest digital sum is 119. It's digital sum is 11 < 14. Hence, we just need to consider only 1 case: The digital sum is 7:

From 100, the next number which has digital sum 5 is 106. Now we have the list:
106, 115, 124 (there are 2 - 0 + 1 = 3 numbers)

The total possible numbers are: 6 + 5 + 3 = 14 numbers.

The answer is C.

EMPOWERgmatRichC · Answer

Hi All,

Based on the answer choices, you know that there are at least 7 numbers that fit the given description, but no more than 20. Thus, listing them all out shouldn't be too difficult. You might also find it useful to organize your numbers by 'group'

We're restricted to the numbers 24 to 125, inclusive. We're asked to find the numbers whose digits add up to a MULTIPLE of 7:

Numbers with digits that add up to 7: 
25 and 52 
34 and 43 
61 
70 
106 
115 
124

Numbers with digits that add up to 14: 
59 and 95 
68 and 86 
77

Total numbers = 9+5 = 14

Final Answer:  C

GMAT assassins aren't born, they're made, 
Rich

soodia · Answer

can someone explain the question, please?
what is it? "The digital sum of a number"

Bunuel · Answer

The digital sum of a number is the sum of its digits. For example, the digital sum of 125 is 1 + 2 + 5 = 8.

Check four different solutions above.

KarishmaB · Answer

It is like a user defined function. It can be given any random name:

"Digital sum" of a number is the sum of its digits.

or

"DSum" of a number is the sum of its digits.

or the question could use any other term. It gives the exact definition of the term and that is what you have to focus on.

gracie · Answer

least integer=25
greatest integer=124
124-25=99
99/7≈14
C

ScottTargetTestPrep · Answer

First, let’s determine the two-digit numbers in the range of 24 to 99, inclusive, that have a digital sum of a multiple of 7:

To get a digital sum of 7, it needs to be 70, 61, 25, 52, 34 or 43.

To get a digital sum of 14, it needs to be 77, 68, 86, 59, or 95.

Since no digital sum of a two-digit number can be 21 or more, we see that there are eleven 2-digit numbers that meet the criteria.

Now, let’s determine the three-digit numbers in the range of 100 to 125, inclusive, that have a digital sum of a multiple of 7:

To get a digital sum of 7, it needs to be 106, 115 or 124.

Since no digital sum of a three-digit number (no more than 125) can be 14 or more, we see that we have a total of three 3-digit numbers that fit the criteria..

Therefore, there are a total of 11 + 3 = 14 such numbers.

Answer: C

gvij2017 · Answer

But when we quote inclusive, we have to add +1. 
So answer must be 15 but because 15 is not an option so we selected 14, isn't it?
Please help!

EMPOWERgmatRichC · Answer

Hi gvij2017,

You're actually referring to a completely different math concept (re: a "fence post problem"). In those types of questions, we have to make sure to include both the 'first item' and the 'last item' in the total. For example, if you read pages 2-5 in a book, you read FOUR pages (not 3).... since you read pages 2, 3, 4 and 5 --> and that's 4 pages. If you subtract 2 from 5, then you only get 5-2 = 3 though (because you're 'removing' page 2 from your total). Thus, you have to 'add it back' (and that's the "+1" that you're referring to).

This question is NOT a fence post problem - it just gives us a range of integers that we have to work within and find the integers that fit a specific description.

GMAT assassins aren't born, they're made,
Rich

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The digital sum of a number is the sum of its digits. For how many of

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