1. **Problem 7b:** Halve the number 48 twice to show it is divisible by 4. - Step 1: Divide 48 by 2: $$48 \div 2 = 24$$ - Step 2: Divide 24 by 2: $$24 \div 2 = 12$$ Since both divisions result in whole numbers, 48 is divisible by 4. 2. **Problem 7c:** Halve the number 84 twice to show it is divisible by 4. - Step 1: Divide 84 by 2: $$84 \div 2 = 42$$ - Step 2: Divide 42 by 2: $$42 \div 2 = 21$$ Since both divisions result in whole numbers, 84 is divisible by 4. 3. **Problem 7d:** Halve the number 124 twice to show it is divisible by 4. - Step 1: Divide 124 by 2: $$124 \div 2 = 62$$ - Step 2: Divide 62 by 2: $$62 \div 2 = 31$$ Since both divisions result in whole numbers, 124 is divisible by 4. 4. **Problem 8b:** Is 1000 divisible by 8? - Step 1: Recall the divisibility rule for 8: A number is divisible by 8 if its last three digits form a number divisible by 8. - Step 2: The last three digits of 1000 are 000, which is 0. - Step 3: Since 0 is divisible by 8, 1000 is divisible by 8. **Note:** The user's original answer said no, but mathematically 1000 is divisible by 8 because $$1000 \div 8 = 125$$ with no remainder. 5. **Problem 9a:** Underline the numbers divisible by 4 from the list: 20, 54, 72, 100, 386, 404, 934, 2480, 17, 392. - Step 1: Check each number's last two digits or divide by 4: - 20: $$20 \div 4 = 5$$ (divisible) - 54: $$54 \div 4 = 13.5$$ (not divisible) - 72: $$72 \div 4 = 18$$ (divisible) - 100: $$100 \div 4 = 25$$ (divisible) - 386: $$386 \div 4 = 96.5$$ (not divisible) - 404: $$404 \div 4 = 101$$ (divisible) - 934: $$934 \div 4 = 233.5$$ (not divisible) - 2480: $$2480 \div 4 = 620$$ (divisible) - 17: $$17 \div 4 = 4.25$$ (not divisible) - 392: $$392 \div 4 = 98$$ (divisible) - Step 2: Underlined numbers divisible by 4 are: 20, 72, 100, 404, 2480, 392. 6. **Problem 9b:** Circle the numbers from part a) that are divisible by 8. - Step 1: Check divisibility by 8 (last three digits divisible by 8 or divide by 8): - 20: $$20 \div 8 = 2.5$$ (no) - 72: $$72 \div 8 = 9$$ (yes) - 100: $$100 \div 8 = 12.5$$ (no) - 404: $$404 \div 8 = 50.5$$ (no) - 2480: $$2480 \div 8 = 310$$ (yes) - 392: $$392 \div 8 = 49$$ (yes) - Step 2: Circled numbers divisible by 8 are: 72, 2480, 392. 7. **Problem 9c:** Sort the numbers into the Venn diagram with circles "Multiples of 4" and "Multiples of 8". - Multiples of 8 are also multiples of 4, so: - Multiples of 8 (and 4): 72, 2480, 392 - Multiples of 4 only (not 8): 20, 100, 404 - Neither: 54, 386, 934, 17 8. **Problem 9d:** Which region of the Venn diagram is empty? - The region for numbers that are multiples of 8 but not multiples of 4 is empty because every multiple of 8 is also a multiple of 4. **Final answers:** - 7b) 48 is divisible by 4 because $$48 \div 2 = 24$$ and $$24 \div 2 = 12$$ are whole numbers. - 7c) 84 is divisible by 4 because $$84 \div 2 = 42$$ and $$42 \div 2 = 21$$ are whole numbers. - 7d) 124 is divisible by 4 because $$124 \div 2 = 62$$ and $$62 \div 2 = 31$$ are whole numbers. - 8b) 1000 is divisible by 8 because $$1000 \div 8 = 125$$ with no remainder. - 9a) Numbers divisible by 4: 20, 72, 100, 404, 2480, 392. - 9b) Numbers divisible by 8: 72, 2480, 392. - 9c) Venn diagram: - Multiples of 8 (and 4): 72, 2480, 392 - Multiples of 4 only: 20, 100, 404 - Neither: 54, 386, 934, 17 - 9d) The region for multiples of 8 but not multiples of 4 is empty.