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Bunuel
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For all positive integers s and t, s@t represents the remainder when 23 Jan 2025, 01:30
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35% (medium)

Question Stats:

76% (01:52) correct 24% (02:17) wrong based on 114 sessions

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For all positive integers s and t, s@t represents the remainder when s is divided by t. What is the sum of all the possible values of t for which 18@t = 2?

A. 38
B. 28
C. 16
D. 12
E. 10


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For all positive integers s and t, s@t represents the remainder when 23 Jan 2025, 02:22
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Bunuel
For all positive integers s and t, s@t represents the remainder when s is divided by t. What is the sum of all the possible values of t for which 18@t = 2?

A. 38
B. 28
C. 16
D. 12
E. 10


­
18%t = 2

t < 18

Factors of 16 (18 - 2), which are not factors of 18 is the required answer

Factors of 16 => 1, 2, 4, 8, 16
Factors of 18 => 1, 2, 3, 6, 9, 18

Factors of 16 which aren't factors of 18 => 4, 8, 16

Sum = 4 + 8 + 16 = 28

IMO: B
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For all positive integers s and t, s@t represents the remainder when 23 Jan 2025, 02:25
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t should be such that when it divides 18 the remainder is 2. Possible values of t -> 4,8 and 16
4+8+16 = 28

Option B
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For all positive integers s and t, s@t represents the remainder when 23 Jan 2025, 02:52
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Step 1: Understand the Problem
The operation s@t gives the remainder when s is divided by t.
Here, 18@t = 2 means the remainder is 2 when 18 is divided by t.
This implies:
18 = qt + 2
where q is an integer quotient, and t > 2 since the remainder is 2.

Step 2: Solve for t
Rearranging the equation:
t = (18 - 2) / q = 16 / q
Here, q must be a factor of 16, as t is an integer.

Step 3: Factors of 16
The positive factors of 16 are 1, 2, 4, 8, 16.
Since t > 2, valid values of t are:
q = 1 → t = 16
q = 2 → t = 8
q = 4 → t = 4

Step 4: Sum of Valid t
t = 4, 8, 16
Sum of t = 4 + 8 + 16 = 28

Final Answer: B. 28
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For all positive integers s and t, s@t represents the remainder when 23 Jan 2025, 07:28
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Given s@t = remainder when s is divided by t
Now the stem asks for all positive integer t, which when dividing 18 gives 2 as remainder.

First let us limit the range of t.
t>0 (as positive integer)
Now when t>18, the remainder always stays 18.
Hence we need to check for 0<t<18

We can quickly see, when t=4 (4*4=16), t=8 (8*2=16) and t= 16 (16*1=16) all will give remainder 2.

Hence, sum of all possible t= 4+8+16 = 28

IMO Ans B
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For all positive integers s and t, s@t represents the remainder when 24 Jan 2025, 22:47
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For 18 the numbers that give reminder 2 are 4 and further multiples of it 8,16.
Hence total = 4+8+16 = 28
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