Divisibility and Factors - ISEE Middle Level: Quantitative Reasoning
Card 1 of 21
What is the definition of a prime number?
What is the definition of a prime number?
Tap to reveal answer
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.
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.
← Didn't Know|Knew It →
What is the definition of a composite number?
What is the definition of a composite number?
Tap to reveal answer
A number $>1$ with more than two positive factors. Composites have factors beyond 1 and themselves, unlike primes or 1.
A number $>1$ with more than two positive factors. Composites have factors beyond 1 and themselves, unlike primes or 1.
← Didn't Know|Knew It →
What is the greatest common factor (GCF) of $18$ and $24$?
What is the greatest common factor (GCF) of $18$ and $24$?
Tap to reveal answer
$6$. Prime factors: 18=2×3², 24=2³×3, so GCF=2×3=6.
$6$. Prime factors: 18=2×3², 24=2³×3, so GCF=2×3=6.
← Didn't Know|Knew It →
What is the least common multiple (LCM) of $6$ and $15$?
What is the least common multiple (LCM) of $6$ and $15$?
Tap to reveal answer
$30$. Prime factors: 6=2×3, 15=3×5, so LCM=2×3×5=30.
$30$. Prime factors: 6=2×3, 15=3×5, so LCM=2×3×5=30.
← Didn't Know|Knew It →
Which statement is always true: If a number is divisible by $9$, is it divisible by $3$?
Which statement is always true: If a number is divisible by $9$, is it divisible by $3$?
Tap to reveal answer
Yes, divisibility by $9$ always implies divisibility by $3$. Since 9=3², any multiple of 9 is also a multiple of 3.
Yes, divisibility by $9$ always implies divisibility by $3$. Since 9=3², any multiple of 9 is also a multiple of 3.
← Didn't Know|Knew It →
Identify the missing digit $x$ so that $45x$ is divisible by $9$.
Identify the missing digit $x$ so that $45x$ is divisible by $9$.
Tap to reveal answer
$x=0$ 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.
$x=0$ 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.
← Didn't Know|Knew It →
What is the prime factorization of $84$?
What is the prime factorization of $84$?
Tap to reveal answer
$2^2 \cdot 3 \cdot 7$. 84=2×42=2×2×21=2²×3×7, the complete prime factorization.
$2^2 \cdot 3 \cdot 7$. 84=2×42=2×2×21=2²×3×7, the complete prime factorization.
← Didn't Know|Knew It →
What is the greatest possible number of positive factors of $p^3$ where $p$ is prime?
What is the greatest possible number of positive factors of $p^3$ where $p$ is prime?
Tap to reveal answer
$4$. For prime p, p³ has factors 1, p, p², p³, totaling 4, the maximum for that form.
$4$. For prime p, p³ has factors 1, p, p², p³, totaling 4, the maximum for that form.
← Didn't Know|Knew It →
What is the divisibility rule for $2$?
What is the divisibility rule for $2$?
Tap to reveal answer
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.
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.
← Didn't Know|Knew It →
What is the divisibility rule for $3$?
What is the divisibility rule for $3$?
Tap to reveal answer
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.
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.
← Didn't Know|Knew It →
What is the divisibility rule for $4$?
What is the divisibility rule for $4$?
Tap to reveal answer
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.
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.
← Didn't Know|Knew It →
What is the divisibility rule for $6$?
What is the divisibility rule for $6$?
Tap to reveal answer
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.
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.
← Didn't Know|Knew It →
What is the divisibility rule for $8$?
What is the divisibility rule for $8$?
Tap to reveal answer
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.
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.
← Didn't Know|Knew It →
What is the divisibility rule for $9$?
What is the divisibility rule for $9$?
Tap to reveal answer
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.
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.
← Didn't Know|Knew It →
What is the divisibility rule for $11$?
What is the divisibility rule for $11$?
Tap to reveal answer
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.
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.
← Didn't Know|Knew It →
Identify whether $378$ is divisible by $3$.
Identify whether $378$ is divisible by $3$.
Tap to reveal answer
Yes, $378$ is divisible by $3$. Sum of digits 3+7+8=18, and 18÷3=6, so divisible by 3.
Yes, $378$ is divisible by $3$. Sum of digits 3+7+8=18, and 18÷3=6, so divisible by 3.
← Didn't Know|Knew It →
Identify whether $7,452$ is divisible by $4$.
Identify whether $7,452$ is divisible by $4$.
Tap to reveal answer
Yes, $7,452$ is divisible by $4$. Last two digits 52, and 52÷4=13, an integer, so divisible by 4.
Yes, $7,452$ is divisible by $4$. Last two digits 52, and 52÷4=13, an integer, so divisible by 4.
← Didn't Know|Knew It →
Identify whether $9,135$ is divisible by $5$.
Identify whether $9,135$ is divisible by $5$.
Tap to reveal answer
Yes, $9,135$ is divisible by $5$. Ends with 5, which satisfies the divisibility rule for 5.
Yes, $9,135$ is divisible by $5$. Ends with 5, which satisfies the divisibility rule for 5.
← Didn't Know|Knew It →
Identify whether $2,514$ is divisible by $6$.
Identify whether $2,514$ is divisible by $6$.
Tap to reveal answer
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.
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.
← Didn't Know|Knew It →
Identify whether $12,488$ is divisible by $8$.
Identify whether $12,488$ is divisible by $8$.
Tap to reveal answer
Yes, $12,488$ is divisible by $8$. Last three digits 488, and 488÷8=61, an integer, so divisible by 8.
Yes, $12,488$ is divisible by $8$. Last three digits 488, and 488÷8=61, an integer, so divisible by 8.
← Didn't Know|Knew It →
Identify whether $4,554$ is divisible by $11$.
Identify whether $4,554$ is divisible by $11$.
Tap to reveal answer
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.
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.
← Didn't Know|Knew It →