Integer Properties and Divisibility - Quantitative Reasoning

1. What is the basic divisibility test for $3$ in base $10$?

   Divisible by $3$ iff the sum of digits is divisible by $3$. This rule works because $10 \equiv 1 \pmod{3}$, so the number modulo 3 equals the digit sum modulo 3.

2. What is the remainder when $2^{10}$ is divided by $3$?

   $1$. $2^{10} = 1024$, and $1024 \div 3 = 341$ with remainder 1.

3. What is the basic divisibility test for $6$ in base $10$?

   Divisible by $6$ iff divisible by $2$ and by $3$. Since 6=2*3 and they are coprime, divisibility requires both conditions.

4. What is the basic divisibility test for $9$ in base $10$?

   Divisible by $9$ iff the sum of digits is divisible by $9$. This follows since $10 \equiv 1 \pmod{9}$, equating the number to its digit sum modulo 9.

5. What is the value of $\gcd(a,b)$ when $a$ and $b$ are relatively prime?

   $\gcd(a,b)=1$. Relatively prime integers share no common divisors other than 1.

6. What is the definition of the greatest common divisor $\gcd(a,b)$ for integers $a$ and $b$ not both $0$?

   The greatest positive integer dividing both $a$ and $b$. It identifies the largest positive integer that divides both without remainder.

7. What is the definition of the least common multiple $\mathrm{lcm}(a,b)$ for nonzero integers $a$ and $b$?

   The least positive integer that is a multiple of both $a$ and $b$. It specifies the smallest positive integer divisible by both $a$ and $b$.

8. What is the remainder when $10^{50}+3$ is divided by $10$?

   $3$. $10^{50}$ is divisible by 10, so adding 3 gives remainder 3.

9. Identify the result: If $a \mid b$ and $a \mid c$, what must be true about $a \mid (b-c)$?

   $a \mid (b-c)$. Divisibility holds for differences of multiples.

10. What is $\mathrm{lcm}(a,0)$ for a nonzero integer $a$?

    $\mathrm{lcm}(a,0)=0$. Using the gcd-lcm relation, it evaluates to 0 for nonzero $a$.

11. What is the basic divisibility test for $10$ in base $10$?

    Divisible by $10$ iff the last digit is $0$. In base 10, ending in 0 means it's a multiple of 10.

12. What is the basic divisibility test for $5$ in base $10$?

    Divisible by $5$ iff the last digit is $0$ or $5$. In base 10, these endings make the number a multiple of 5.

13. What is the relationship between $\gcd(a,b)$ and $\mathrm{lcm}(a,b)$ for nonzero integers $a$ and $b$?

    $\gcd(a,b)\,\mathrm{lcm}(a,b)=|ab|$. This formula relates gcd and lcm through the absolute product of the integers.

14. What is $\gcd(a,0)$ for a nonzero integer $a$?

    $\gcd(a,0)=|a|$. Every integer divides 0, so the gcd is the absolute value of $a$.

15. What is the remainder when $7^{100}$ is divided by $7$?

    $0$. Any positive power of 7 is divisible by 7, leaving no remainder.

16. Identify the result: If $a \mid b$, what must be true about $a \mid (bk)$ for any integer $k$?

    $a \mid (bk)$ for every integer $k$. Scaling a multiple by any integer $k$ retains divisibility by $a$.

17. Identify whether $357$ is divisible by $3$.

    Yes, $357$ is divisible by $3$. Digit sum $3+5+7=15$ is divisible by 3, confirming divisibility.

18. What is the basic divisibility test for $4$ in base $10$?

    Divisible by $4$ iff the last two digits form a multiple of $4$. The last two digits represent the number modulo 100, and divisibility by 4 checks modulo 4.

19. What does it mean to say that $a$ is divisible by $b$ (with $b \ne 0$) in the integers?

    $b \mid a$ means $a=bk$ for some integer $k$. This defines divisibility where $a$ is an integer multiple of $b$.

20. What is the basic divisibility test for $2$ in base $10$?

    An integer is divisible by $2$ iff its last digit is even. In base 10, even last digits ensure the number is even.

21. What is the basic divisibility test for $12$ in base $10$?

    Divisible by $12$ iff divisible by $3$ and by $4$. 12=3*4 with coprime factors, so both divisibility tests must hold.

22. What is the basic divisibility test for $8$ in base $10$?

    Divisible by $8$ iff the last three digits form a multiple of $8$. The last three digits form the number modulo 1000, sufficient for divisibility by 8.

23. Identify the result: If $a \mid b$ and $a \mid c$, what must be true about $a \mid (b+c)$?

    $a \mid (b+c)$. Divisibility preserves under addition of multiples.

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