Solutions: Section 5.5

Page 124: Figure it Out

Q. Is 8536 divisible by 4?

Check last two digits: 36. $36 \div 4 = 9$. Answer: Yes.

Page 125: Figure it Out

Q1. Leap Years

b. Leap years from 2024 to 2099. Sequence: 2024, 2028, 2032… This is an Arithmetic Progression with diff 4. $2096$ is the last one (divisible by 4). Count: $(2096 - 2024) \div 4 + 1 = 72 \div 4 + 1 = 18 + 1 = 19$. Answer: 19 leap years.

Q2. Largest/Smallest 4-digit palindrome divisible by 4.

Divisibility by 4: Last two digits must be div by 4. Palindrome: ABBA. So the number ends in A. Since it’s even (div by 4), A must be 2, 4, 6, 8 (cannot be 0 as it’s the first digit too).

Largest: Start with A=8 (Since 9 is odd). 8BB8. Maximize B $\to$ 9. 8998. Div by 4? Last digits 98. $98 \div 4 = 24.5$ (No). Try next B $\to$ 8. 8888. Div by 4? 88 yes. Wait, can A be 6? 6996. $96 \div 4 = 24$. Yes. Is there a larger one starting with 8? 8888 works. Is there one starting with 9? No, must be even. Let’s re-evaluate A. Number: $1000A + 100B + 10B + A$. Ends in $BA$. $10B + A$ divisible by 4. A is even.

Largest starts with 8? 8998 (No). 8888 (Yes). But what if we start with 6? 6996? Wait, A must be the leading digit. 8 is larger than 6. So 8888 is a candidate. Try A=2, 4, 6, 8. Let’s try finding a 9… No, must be even. Is there a number beginning with 9? 9xx9. Ends in 9 (Odd). Not divisible by 4. So max digit is 8. Candidate: 8998 (No). 8888 (Yes). What about B? We want max B. Try 8998 (98 not div by 4). Try 8778 (78 not div by 4). The condition for $10B + A$ divisible by 4 (where A=8): $10B + 8$. Since 8 is div by 4, $10B$ must be div by 4. $10B = 2 \times 5 \times B$. Needs another factor of 2. So B must be even. Max B = 8. Largest: 8888? Wait, check A=6. 6996. (96 div by 4). But 6 < 8. Check A=4. 4994 (94 no). Check A=2. 2992 (92 yes).

Wait, I missed that 9000s are not possible. So Largest is 8888.

Smallest: Start with A=2 (A=0 impossible). 2002. 02 not div by 4. 2112. 12 div by 4. Yes. Can we go lower? Start with 1? No, 1 is odd. Wait, A=1 is impossible for div by 4. Start with A=2? 2002 (No). 2112 (Yes). Is there a 4-digit number starting with 1 that is a palindrome and div by 4? 1xx1 $\to$ Ends in 1 $\to$ Odd $\to$ No. So smallest must start with 2. Smallest B=0? 2002 (No). Smallest B=1? 2112. Answer: Largest 8888, Smallest 2112. Correction from PDF Solution Check: PDF Solution says Smallest: 1001 (Wait, 1001 is not div by 4, it’s odd). PDF Solution Page 32 says: “Smallest 4-digit number divisible by 4 and also a palindrome: 1001.” CRITICAL ERROR IN SOURCE PDF: 1001 is odd. It cannot be divisible by 4. My logic: Smallest must be even. Let’s check the PDF text again. Q2: “Find the largest and smallest 4-digit numbers that are divisible by 4 and are also palindromes.” Solution in PDF (Page 32) explicitly says “1001”. This is mathematically incorrect ($1001 \div 4 = 250.25$). However, as an AI, I should point this out but maybe provide the mathematically correct one or note the discrepancy. Actually, let’s look at the largest in PDF: 9999. ($9999$ is odd too!). The PDF solution page 32 seems to have ignored the “divisible by 4” constraint for the answer or provided a wrong answer. Wait, let me re-read the PDF image. PDF Page 32 Q2 Ans: “The largest 4-digit number divisible by 4 and also a palindrome: 9999… The smallest… 1001”. Both are odd. Constraint: “Divisible by 4”. This is a hallucination in the PDF’s answer key or a typo in the question in the PDF (maybe they meant divisible by something else?). Given my role is to generate “high-quality, exam-oriented study material”, I should provide the correct mathematical answer and perhaps add a note. Correct answers: Largest: 8888. Smallest: 2112. Actually, let’s re-verify A=1… No. Let’s check A=2, B=1 -> 2112 / 4 = 528. Yes. Let’s check A=2, B=0 -> 2002 / 4 = 500.5. No. I will provide the mathematically correct calculation.

Page 126

Q4. Remainders

Number: 78

• Div by 10: Remainder 8.
• Div by 5: $75$ is multiple. Remainder $78-75=3$.
• Div by 2: Even. Remainder 0.

Q5. 14560 divisibility

Teacher asks 2, 4, 5, 8, 10. Guna checked two. Which two imply the rest?

• If div by 10 $\to$ div by 2 and 5.
• If div by 8 $\to$ div by 2 and 4. So checking 8 and 10 covers everything.
• 10 covers 5, 2.
• 8 covers 4, 2.
• Combined: 2, 4, 5, 8, 10. Answer: 8 and 10.