Divisibility and Factors - ISEE Lower Level Quantitative Reasoning

What is the remainder when $53$ is divided by $5$ ?

$3$ . 53 ÷ 5 = 10 with remainder 3, as 5 × 10 = 50 and 53 - 50 = 3.
What is the least common multiple (LCM) of two positive integers?

The smallest positive integer that is a multiple of both integers. This is the minimal number encompassing all prime factors of both at their highest powers.
What is the divisibility rule for $9$ ?

Divisible by $9$ if the sum of digits is divisible by $9$ . This is due to the number being congruent to the sum of its digits modulo 9.
What is the greatest common factor (GCF) of two integers?

The greatest positive integer that divides both integers. This measures the largest shared divisor, found via prime factorization or Euclidean algorithm.
Which option is a factor of $84$ : $5$ , $6$ , or $11$ ?

$6$ . 84 ÷ 6 = 14 (integer), while 84 ÷ 5 = 16.8 and 84 ÷ 11 ≈ 7.636 are not integers.
What is the divisibility rule for $10$ ?

Divisible by $10$ if the last digit is $0$ . This rule holds because multiples of 10 in base 10 must end in 0.
Identify the LCM of $6$ and $8$ .

$24$ . Prime factors: 6 = 2 × 3, 8 = $2^3$ ; LCM takes maximum exponents: $2^3$ × 3 = 24.
Identify whether $7{,}152$ is divisible by $8$ .

Yes, $7{,}152$ is divisible by $8$ . Last three digits: 152 ÷ 8 = 19 (integer), confirming divisibility by 8.
What is the prime factorization of $60$ ?

$2^2 \cdot 3 \cdot 5$ . Decompose 60 as 2 × 30, then 2 × 2 × 15, and 3 × 5, yielding its prime factors.
Identify the GCF of $24$ and $36$ .

$12$ . Prime factors: 24 = $2^3$ × 3, 36 = $2^2$ × $3^2$ ; GCF takes minimum exponents: $2^2$ × 3 = 12.
Identify the greatest integer that divides both $48$ and $72$ .

$24$ . This is the GCF of 48 ( $2^4$ × 3) and 72 ( $2^3$ × $3^2$ ), using min exponents: $2^3$ × 3 = 24.
What is the greatest prime factor of $45$ ?

$5$ . Factorize 45 = $3^2$ × 5; the prime factors are 3 and 5, with 5 being the largest.
What is the divisibility rule for $2$ ?

A number is divisible by $2$ if its last digit is even. This rule holds because in base 10, the number's evenness depends solely on its units digit being even.
Identify whether $231$ is divisible by $3$ .

Yes, $231$ is divisible by $3$ . Sum of digits: 2 + 3 + 1 = 6, which is divisible by 3, so 231 is divisible by 3.
What is a composite number?

A whole number greater than $1$ with more than two positive factors. This distinguishes numbers that are not prime, having factors beyond 1 and themselves.
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 both ensures divisibility by their product.
What is the smallest positive integer divisible by both $9$ and $12$ ?

$36$ . This is the LCM of 9 ( $3^2$ ) and 12 ( $2^2$ × 3), taking max exponents: $2^2$ × $3^2$ = 36.
What is the divisibility rule for $3$ ?

Divisible by $3$ if the sum of digits is divisible by $3$ . This stems from the fact that a number is congruent to the sum of its digits modulo 3.
What is the divisibility rule for $8$ ?

Divisible by $8$ if the last $3$ digits form a multiple of $8$ . This checks the number modulo 1000, as 1000 is divisible by 8, confirming divisibility by 8.
What is a prime number?

A whole number greater than $1$ with exactly two positive factors. This definition identifies numbers only divisible by 1 and themselves, with no other factors.
Identify whether $1{,}248$ is divisible by $4$ .

Yes, $1{,}248$ is divisible by $4$ . Last two digits: 48 ÷ 4 = 12 (integer), confirming divisibility by 4.
What is the divisibility rule for $5$ ?

Divisible by $5$ if the last digit is $0$ or $5$ . This rule applies because multiples of 5 in base 10 end in 0 or 5.
What is the divisibility rule for $4$ ?

Divisible by $4$ if the last $2$ digits form a multiple of $4$ . This works as the last two digits represent the number modulo 100, and divisibility by 4 checks modulo 4.

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