Divisibility and Factors - ISEE Middle Level Quantitative Reasoning

What is the definition of a prime number?

A number $>1$ with exactly two positive factors: $1$ and itself. Primes have no positive divisors other than 1 and themselves, distinguishing them from 1 and composites.
What is the greatest possible number of positive factors of $p^3$ where $p$ is prime?

$4$ . For prime p, p³ has factors 1, p, p², p³, totaling 4, the maximum for that form.
What is the divisibility rule for $2$ ?

A number is divisible by $2$ if its last digit is even. A number's parity is determined solely by its last digit, so if even, the whole number is even and divisible by 2.
What is the greatest common factor (GCF) of $18$ and $24$ ?

$6$ . Prime factors: 18=2×3², 24=2³×3, so GCF=2×3=6.
Identify whether $9,135$ is divisible by $5$ .

Yes, $9,135$ is divisible by $5$ . Ends with 5, which satisfies the divisibility rule for 5.
Identify whether $378$ is divisible by $3$ .

Yes, $378$ is divisible by $3$ . Sum of digits 3+7+8=18, and 18÷3=6, so divisible by 3.
What is the least common multiple (LCM) of $6$ and $15$ ?

$30$ . Prime factors: 6=2×3, 15=3×5, so LCM=2×3×5=30.
Identify whether $2,514$ is divisible by $6$ .

Yes, $2,514$ is divisible by $6$ . Even (ends with 4) and sum 2+5+1+4=12 divisible by 3, so divisible by 6.
What is the divisibility rule for $4$ ?

Divisible by $4$ if the last two digits form a multiple of $4$ . The last two digits represent the number modulo 100, and since 100 is divisible by 4, divisibility depends on those digits.
What is the divisibility rule for $6$ ?

Divisible by $6$ if divisible by both $2$ and $3$ . Since 6=2×3 and 2 and 3 are coprime, divisibility by 6 requires satisfying both rules.
Identify the missing digit $x$ so that $45x$ is divisible by $9$ .

$x=0$ or $x=9$ . Sum of digits 4+5+x=9+x must be divisible by 9, so for x=0-9, 9 or 18, thus x=0 or 9.
What is the divisibility rule for $8$ ?

Divisible by $8$ if the last three digits form a multiple of $8$ . The last three digits form the number modulo 1000, and 1000=8×125, so divisibility by 8 depends on those digits.
What is the definition of a composite number?

A number $>1$ with more than two positive factors. Composites have factors beyond 1 and themselves, unlike primes or 1.
Identify whether $7,452$ is divisible by $4$ .

Yes, $7,452$ is divisible by $4$ . Last two digits 52, and 52÷4=13, an integer, so divisible by 4.
What is the divisibility rule for $11$ ?

Divisible by $11$ if the alternating digit sum is a multiple of $11$ . The alternating sum is equivalent to the number modulo 11, since 10≡-1 mod 11, alternating powers of -1.
What is the prime factorization of $84$ ?

$2^2 \cdot 3 \cdot 7$ . 84=2×42=2×2×21=2²×3×7, the complete prime factorization.
Identify whether $12,488$ is divisible by $8$ .

Yes, $12,488$ is divisible by $8$ . Last three digits 488, and 488÷8=61, an integer, so divisible by 8.
Which statement is always true: If a number is divisible by $9$ , is it divisible by $3$ ?

Yes, divisibility by $9$ always implies divisibility by $3$ . Since 9=3², any multiple of 9 is also a multiple of 3.
What is the divisibility rule for $3$ ?

Divisible by $3$ if the sum of digits is divisible by $3$ . A number is congruent to the sum of its digits modulo 3, so if the sum is divisible by 3, the number is too.
Identify whether $4,554$ is divisible by $11$ .

Yes, $4,554$ is divisible by $11$ . Alternating sum 4-5+5-4=0, and 0 is a multiple of 11, so divisible by 11.
What is the divisibility rule for $9$ ?

Divisible by $9$ if the sum of digits is divisible by $9$ . A number is congruent to the sum of its digits modulo 9, so if the sum is divisible by 9, the number is too.

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