Bunuel wrote:

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

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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