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

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The digital sum of a number is the sum of its digits. For how many of [#permalink]

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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
[Reveal] Spoiler: OA

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Re: The digital sum of a number is the sum of its digits. For how many of [#permalink]

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is there other way than just listing?

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

14 ways..

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Re: The digital sum of a number is the sum of its digits. For how many of [#permalink]

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mmelendez wrote:
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



Kudos if you like this question and the solution I will provide

:-D


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)
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Re: The digital sum of a number is the sum of its digits. For how many of [#permalink]

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mmelendez wrote:
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

Kudos if you like this question and the solution I will provide

:-D


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.
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Re: The digital sum of a number is the sum of its digits. For how many of [#permalink]

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New post 25 Nov 2017, 13:46
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:
[Reveal] Spoiler:
C


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Re: The digital sum of a number is the sum of its digits. For how many of [#permalink]

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New post 27 Nov 2017, 02:20
can someone explain the question, please?
what is it? "The digital sum of a number"

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Re: The digital sum of a number is the sum of its digits. For how many of [#permalink]

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Re: The digital sum of a number is the sum of its digits. For how many of [#permalink]

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New post 28 Nov 2017, 02:39
soodia wrote:
can someone explain the question, please?
what is it? "The digital sum of a number"


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.
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The digital sum of a number is the sum of its digits. For how many of [#permalink]

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New post 28 Nov 2017, 20:56
mmelendez wrote:
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


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

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The digital sum of a number is the sum of its digits. For how many of   [#permalink] 28 Nov 2017, 20:56
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