Understanding Divisibility by 22

To find which number is divisible by 22, we need to check if the number satisfies the divisibility rules for both 2 and 11. This is because 22 is the product of 2 and 11, which are prime numbers.

Key Divisibility Rules

• Rule for 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).
• Rule for 11: A number is divisible by 11 if the alternating sum of its digits (starting from the left) is either 0 or a multiple of 11. The calculation is: (Sum of digits at odd places) - (Sum of digits at even places).

Analyzing the Options

Option 1: 7986

Divisibility by 2: The last digit is 6, which is even. So, 7986 is divisible by 2.

Divisibility by 11:

• Digits: 7, 9, 8, 6
• Sum of digits at odd places (1st digit, 3rd digit): $7 + 8 = 15$
• Sum of digits at even places (2nd digit, 4th digit): $9 + 6 = 15$
• Difference: $15 - 15 = 0$

Since the difference is 0, 7986 is divisible by 11.

Result: Because 7986 is divisible by both 2 and 11, it is divisible by 22.

Option 2: 7019

Divisibility by 2: The last digit is 9, which is odd. So, 7019 is not divisible by 2.

Result: Since it's not divisible by 2, 7019 is not divisible by 22.

Option 3: 7664

Divisibility by 2: The last digit is 4, which is even. So, 7664 is divisible by 2.

Divisibility by 11:

• Digits: 7, 6, 6, 4
• Sum of digits at odd places (1st and 3rd): $7 + 6 = 13$
• Sum of digits at even places (2nd and 4th): $6 + 4 = 10$
• Difference: $13 - 10 = 3$

Since the difference is 3 (not 0 or a multiple of 11), 7664 is not divisible by 11.

Result: Since it's not divisible by 11, 7664 is not divisible by 22.

Option 4: 8371

Divisibility by 2: The last digit is 1, which is odd. So, 8371 is not divisible by 2.

Result: Since it's not divisible by 2, 8371 is not divisible by 22.

Conclusion

By applying the divisibility rules for 2 and 11, we found that only the number 7986 is divisible by both, making it the correct answer for divisibility by 22.