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Bunuel
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If k is the sum of the digits of integer m, and m=18n, where n is an 09 Apr 2021, 13:00
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If k is the sum of the digits of integer m, and m=18n, where n is an integer, which of the following must be true?

A. The sum of the digits of m is 9
B. The sum of the digits of k is 9
C. m is a multiple of 2k
D. k is a multiple of 9
E. k is a multiple of 6
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D
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If k is the sum of the digits of integer m, and m=18n, where n is an 09 Apr 2021, 13:31
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Take numbers for m, n. Let n=1,2,3... for any value of n, the sum of the digits of M is a multiple of 9.

Therefore k is a multiple of 9
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If k is the sum of the digits of integer m, and m=18n, where n is an 09 Apr 2021, 14:24
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I understand that K is a multiple of 9, but how is A wrong? For every value of n multiplied by 18 is also a multiple of 18 and therefore a multiple of 9.
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If k is the sum of the digits of integer m, and m=18n, where n is an 10 Apr 2021, 14:49
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ebear444
I understand that K is a multiple of 9, but how is A wrong? For every value of n multiplied by 18 is also a multiple of 18 and therefore a multiple of 9.

Take e.g. n=11
this is why A is wrong

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If k is the sum of the digits of integer m, and m=18n, where n is an 11 Apr 2021, 08:35
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ebear444
I understand that K is a multiple of 9, but how is A wrong? For every value of n multiplied by 18 is also a multiple of 18 and therefore a multiple of 9.
A cannot be true because the question says that the value of "n" can be an integer. We all know that 0 is an integer. Hence in that specific case, the value of k is also going to be 0. Hence we can eliminate A.
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If k is the sum of the digits of integer m, and m=18n, where n is an 11 Apr 2021, 10:43
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If k is the sum of the digits of integer m, and m=18n, where n is an integer, which of the following must be true?

A. The sum of the digits of m is 9
B. The sum of the digits of k is 9
C. m is a multiple of 2k
D. k is a multiple of 9
E. k is a multiple of 6


m=18n, n = 1,2,3,...........................
n= 1, m = 18, k= 1+8=9
n=2, m= 36, k=6+3=9
.......
n=55, m= 990, k= 9+9=9*2

so , k is a multiple of 9

Answer D
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If k is the sum of the digits of integer m, and m=18n, where n is an 30 Jul 2023, 09:51
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Asked: If k is the sum of the digits of integer m, and m=18n, where n is an integer, which of the following must be true?

m = 18; k = 9
m = 36; k = 9
m = 54; k = 9
m = 72; k = 9
m = 90; k = 9
m = 108; k = 9
m = 126; k = 9
m = 144; k = 9
m = 162; k = 9
m = 180; k = 9
m = 198; k = 18

k is a multiple of 9

IMO D
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If k is the sum of the digits of integer m, and m=18n, where n is an 30 Jul 2023, 10:19
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Bunuel
If k is the sum of the digits of integer m, and m=18n, where n is an integer, which of the following must be true?

A. The sum of the digits of m is 9
B. The sum of the digits of k is 9
C. m is a multiple of 2k
D. k is a multiple of 9
E. k is a multiple of 6


Since m is a multiple of 18, it must be a multiple of 9.
Since m is divisible by 9, the sum of digits of m, i.e., k, must be divisible by 9 (divisibility rule of 9)

The reason for this rule is:
Any number, say, of 3 digits a, b and c can be written as 100a + 10b + c = 99a + 9b + (a + b + c)
Since 99a + 9b is surely divisible by 9, we just need to check the divisibility of (a + b + c), i.e., the sum of digits

Answer D
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If k is the sum of the digits of integer m, and m=18n, where n is an 31 Jul 2023, 13:36
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Can anybody explain why B is wrong?
18*2=36
Then K=9
Sum of digits of K is 9
18*102=1836
Then K=18
Sum of digits of K is again 9

Then how come B true. I understand the logic behind D being the answer.

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If k is the sum of the digits of integer m, and m=18n, where n is an 31 Jul 2023, 21:43
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Mapat
Can anybody explain why B is wrong?
18*2=36
Then K=9
Sum of digits of K is 9
18*102=1836
Then K=18
Sum of digits of K is again 9

Then how come B true. I understand the logic behind D being the answer.

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The sum of the digits of k is not necessarily 9. For example, if m = 18*11,111,111,111 = 199,999,999,998, then k = 99 and the sum of the digits of k is 18, not 9.
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If k is the sum of the digits of integer m, and m=18n, where n is an 09 Jan 2025, 05:21
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