14:I[5014,["/_next/static/chunks/11d1gk4m2o846.js","/_next/static/chunks/14mrh2-p_w84d.js","/_next/static/chunks/40y9lkvta2o_p.js"],"default"] 15:I[38392,["/_next/static/chunks/11d1gk4m2o846.js","/_next/static/chunks/14mrh2-p_w84d.js","/_next/static/chunks/40y9lkvta2o_p.js"],"PracticeCard"] 17:I[70570,["/_next/static/chunks/11d1gk4m2o846.js","/_next/static/chunks/14mrh2-p_w84d.js","/_next/static/chunks/40y9lkvta2o_p.js"],"ExamAppearanceBadge"] 16:T4d7,Question: When x is divided by 7, the remainder is 6. Which of the following must be an even number?

Solution:
Given that when x is divided by 7, the remainder is 6, we can express x as:
x = 7k + 6 where k is an integer.

Now, let's evaluate each option to see which, if any, must be an even number.

Option ক)
x + 6
x + 6 = (7k + 6) + 6 = 7k +12
Since 7k is odd or even depending on k, adding 12 (which is even) does not guarantee that x + 6 will be even.
Therefore, x + 6 is not necessarily even.

Option খ)
x² + x
x² + x = (7k+6)² + (7k+6) Let's expand and simplify:
x² + x = (49k² + 84k + 36) + (7k + 6) = 49k² + 91k + 42
All terms are multiples of 7, and 42 is an even number, making the entire expression even.
Thus, x² + x must be an even number.

Option গ)
x - 4
x - 4 = (7k + 6) - 4 = 7k + 2
This is not necessarily even because 7k + 2 depends on the parity of k.
Therefore, x - 4 is not necessarily even.

Conclusion:
The correct answer is খ) x² + x