1. **State the problem:** Determine whether each number (88, 48, 22, 132, 264) is divisible by 4, 8, 11, and 12. 2. **Recall divisibility rules:** - Divisible by 4: The last two digits form a number divisible by 4. - Divisible by 8: The last three digits form a number divisible by 8. - Divisible by 11: The difference between the sum of digits in odd positions and the sum of digits in even positions is 0 or a multiple of 11. - Divisible by 12: The number is divisible by both 3 and 4. 3. **Check each number:** **1) 88** - Last two digits: 88, divisible by 4 since $88 \div 4 = 22$. - Last three digits: 88 (only two digits), check 8 divisibility: $88 \div 8 = 11$, divisible. - Sum of digits: $8 + 8 = 16$, divisible by 3? No. - Divisible by 12? No, since not divisible by 3. - For 11: Odd positions digits = 8 (1st), Even positions digits = 8 (2nd), difference $8 - 8 = 0$, divisible by 11. **2) 48** - Last two digits: 48, divisible by 4 since $48 \div 4 = 12$. - Last three digits: 48, check 8 divisibility: $48 \div 8 = 6$, divisible. - Sum of digits: $4 + 8 = 12$, divisible by 3. - Divisible by 12? Yes, divisible by 3 and 4. - For 11: Odd positions digits = 4, Even positions digits = 8, difference $4 - 8 = -4$, not divisible by 11. **3) 22** - Last two digits: 22, check divisibility by 4: $22 \div 4 = 5.5$, not divisible. - Last three digits: 22, check divisibility by 8: $22 \div 8 = 2.75$, not divisible. - Sum of digits: $2 + 2 = 4$, not divisible by 3. - Divisible by 12? No. - For 11: Odd positions digits = 2, Even positions digits = 2, difference $2 - 2 = 0$, divisible by 11. **4) 132** - Last two digits: 32, divisible by 4 since $32 \div 4 = 8$. - Last three digits: 132, check divisibility by 8: $132 \div 8 = 16.5$, not divisible. - Sum of digits: $1 + 3 + 2 = 6$, divisible by 3. - Divisible by 12? No, since not divisible by 8. - For 11: Odd positions digits = 1 + 2 = 3, Even positions digits = 3, difference $3 - 3 = 0$, divisible by 11. **5) 264** - Last two digits: 64, divisible by 4 since $64 \div 4 = 16$. - Last three digits: 264, check divisibility by 8: $264 \div 8 = 33$, divisible. - Sum of digits: $2 + 6 + 4 = 12$, divisible by 3. - Divisible by 12? Yes, divisible by 3 and 4. - For 11: Odd positions digits = 2 + 4 = 6, Even positions digits = 6, difference $6 - 6 = 0$, divisible by 11. 4. **Summary table with checks:** | Number | 4 | 8 | 12 | 11 | |--------|---|---|----|----| | 88 | ✓ | ✓ | | ✓ | | 48 | ✓ | ✓ | ✓ | | | 22 | | | | ✓ | | 132 | ✓ | | | ✓ | | 264 | ✓ | ✓ | ✓ | ✓ | **Final answers:** - 88 is divisible by 4, 8, and 11. - 48 is divisible by 4, 8, and 12. - 22 is divisible by 11 only. - 132 is divisible by 4 and 11. - 264 is divisible by 4, 8, 11, and 12.