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When the integer k is divided by 7, the remainder is 5. Which of the 04 Dec 2014, 07:40
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E
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When the integer k is divided by 7, the remainder is 5. Which of the following expressions below when divided by 7, will have a remainder of 6?

I. 4k + 7
II. 6k + 4
III. 8k + 1

A) I only
B) II only
C) III only
D) I and II only
E) I, II and III


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E
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When the integer k is divided by 7, the remainder is 5. Which of the 04 Dec 2014, 11:14
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Bunuel

Tough and Tricky questions: Remainders.



When the integer k is divided by 7, the remainder is 5. Which of the following expressions below when divided by 7, will have a remainder of 6?

I. 4k + 7
II. 6k + 4
III. 8k + 1

A) I only
B) II only
C) III only
D) I and II only
E) I, II and III

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k= 7t+5 where t can be 0,1,2,----
I. 4k+7= 4(7t+5) + 7 = 28t + 27
28t is completely divisible by 7. thus remainder when 28t+27 is divided by 7 = 27/7 = 6

II. 6k+4= 6(7t+5) +4 = 42t+34
again 42t is completely divisible by 7. thus remainder when 42t+34 is divided by 7 = 34/7 = 6

II. 8k+1 = 8(7t+5)+1 = 56t+41
again by using similar analogy remainder = 41/7 =6

now, since each I,II and III leaves a remainder of 6. hence answer must be E
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When the integer k is divided by 7, the remainder is 5. Which of the 04 Dec 2014, 10:57
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The options for the integer k are 5, 12, 19, 26...

Let's look at each of the three cases:
1. 4k+7 yields 27, 55... yes
2. 6k + 4 yields 34, 76... yes
3. 8k + 1 yields 41, 97... yes

I didn't try anymore and concluded that they all work. Choice E.
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When the integer k is divided by 7, the remainder is 5. Which of the 05 Dec 2014, 00:12
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Answer = E) I, II and III

Place k = 12

I: 4(12) + 7 = 55 = 56-1 >> Remainder = 6

II: 6(12) + 4 = 72 + 4 = 76 = (77-1) >> Remainder = 6

III: 8(12) + 1 = 96 + 1 = 97 = (96-1) >> Remainder = 6
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When the integer k is divided by 7, the remainder is 5. Which of the 11 Aug 2019, 19:18
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Bunuel

Tough and Tricky questions: Remainders.



When the integer k is divided by 7, the remainder is 5. Which of the following expressions below when divided by 7, will have a remainder of 6?

I. 4k + 7
II. 6k + 4
III. 8k + 1

A) I only
B) II only
C) III only
D) I and II only
E) I, II and III

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k can be values such as 5, 12, 19, 26, etc.

Let’s test the answers using 5 and 12 as the values of k.

I. 4k + 7

When k = 5, 4k + 7 is 27, which does have a remainder of 6 when divided by 7.

When k = 12, 4k + 7 is 55, which does have a remainder of 6 when divided by 7

We can keep testing values but we will get 6 as the remainder. So I is true.

II. 6k + 4

When k = 5, 6k + 4 is 34, which does have a remainder of 6 when divided by 7.

When k = 12, 6k + 4 is 76, which does have a remainder of 6 when divided by 7

We can keep testing values, but we will get 6 as the remainder. So II is true.

III. 8k + 1

When k = 5, 8k + 1 is 41, which does have a remainder of 6 when divided by 7.

When k = 12, 8k + 1 is 97, which does have a remainder of 6 when divided by 7

We can keep testing values, but we will get 6 as the remainder. So III is true.

Alternate Solution:

Let’s note the following fact: If the remainder when n is divided by 7 is 6, then n + 1 is divisible by 7.

Since k produces a remainder of 5 when divided by 7, we can write k = 7s + 5 for some positive integer s. We will test each Roman numeral, but rather than testing whether the remainder is 6, we will use the above fact and test whether one more than that expression is divisible by 7.

I. 4k + 7

Let’s test whether 4k + 8 is divisible by 7. Substituting k = 7s + 5, we get:

4k + 8 = 4(7s + 5) + 8 = 28s + 28.

Since 28s + 28 is divisible by 7, 4k + 7 will produce a remainder of 6 when divided by 7.

II. 6k + 4

Let’s test whether 6k + 5 is divisible by 7. Substituting k = 7s + 5, we get:

6k + 5 = 6(7s + 5) + 5 = 42s + 35

Since 42s + 35 is divisible by 7, 6k + 4 will produce a remainder of 6 when divided by 7.

III. 8k + 1

Let’s test whether 8k + 2 is divisible by 7. Substituting k = 7s + 5, we get:

8k + 2 = 8(7s + 5) + 2 = 56s + 42

Since 56s + 42 is divisible by 7, 8k + 1 will produce a remainder of 6 when divided by 7.

Answer: E
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When the integer k is divided by 7, the remainder is 5. Which of the 10 Jan 2026, 10:22
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