Numbers that leave a remainder of 2 when divided by 7 can be written in the form:a)7k + 1b)7k + 2c)7k + 3d)7k – 2Correct answer is option 'B'. Can you explain this answer?

Question

Answer

1) Division algorithm (the rule behind remainders)

For any whole number n and any positive divisor 7, there exist integers k (the quotient) and r (the remainder) such that
n = 7k + r, with 0 ≤ r ≤ 6.

So the possible remainders on division by 7 are only 0, 1, 2, 3, 4, 5, 6.

2) Apply it to “remainder = 2”

If a number leaves remainder 2 when divided by 7, then r = 2 in the formula:
n = 7k + 2.
This is exactly option (b) 7k + 2.

3) Quick checks with examples

Take k = 0 → n = 7·0 + 2 = 2 → 2 ÷ 7 = 0 remainder 2
k = 1 → n = 9 → 9 ÷ 7 = 1 remainder 2
k = 2 → n = 16 → 16 ÷ 7 = 2 remainder 2
k = 5 → n = 37 → 37 ÷ 7 = 5 remainder 2
So all numbers of the form 7k + 2 give remainder 2.

4) Why the other options are wrong

(a) 7k + 1 → remainder is 1 (example: 8 ÷ 7 leaves 1).
(c) 7k + 3 → remainder is 3 (example: 10 ÷ 7 leaves 3).
(d) 7k – 2 → this looks tempting, but note:
7k – 2 = 7(k – 1) + 5, so the standard (non-negative) remainder is 5, not 2.
Example: k = 2 → 7k – 2 = 12, and 12 ÷ 7 leaves remainder 5.

5) One-line “test”

To see which form matches remainder r, just reduce it mod 7:

7k + r ≡ r (mod 7).
Only 7k + 2 is ≡ 2 (mod 7).

Therefore, numbers that leave remainder 2 when divided by 7 are exactly those of the form 7k + 2 (option b).

Answer

Understanding Remainders
When a number is divided by another number, the remainder is what is left over after division. For example, if we divide 9 by 7, the quotient is 1 and the remainder is 2.
Formulating the Problem
In this case, we are looking for numbers that leave a remainder of 2 when divided by 7. We can express these numbers mathematically.
General Form of Numbers
If we denote any integer by 'k', then:
- When a number is divided by 7, it can be expressed in the form of 7k plus some remainder.
- Since we want a remainder of 2, we can write the expression as:
- 7k + 2
This means that when we divide this expression by 7, the remainder will always be 2.
Evaluating the Options
Let's evaluate the given options to find which one matches the requirement:
- Option a: 7k + 1 - This leaves a remainder of 1 when divided by 7.
- Option b: 7k + 2 - This leaves a remainder of 2 when divided by 7. (Correct Answer)
- Option c: 7k + 3 - This leaves a remainder of 3 when divided by 7.
- Option d: 7k - 2 - This can be rewritten as 7(k-1) + 5, which leaves a remainder of 5 when divided by 7.
Conclusion
The only expression that fits our requirement of leaving a remainder of 2 when divided by 7 is 7k + 2 (option B). This correctly identifies that all such numbers can be represented in this form.

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