Divisibility Rules - SSAT Upper Level Quantitative

Identify whether $13,530$ is divisible by $10$ .

Yes, because the last digit is $0$ . The rule for 10 is met as the number ends in 0, confirming divisibility.
State the divisibility rule for $5$ .

Divisible by $5$ iff the last digit is $0$ or $5$ . Since $10 \equiv 0 \pmod{5}$ , the number is divisible by 5 precisely when its units digit is 0 or 5.
Identify whether $10^6-1$ is divisible by $9$ .

Yes, $10^6-1=999999$ and $9+9+9+9+9+9=54$ is divisible by $9$ . The sum of digits 54 is divisible by 9 ( $54 \div 9 = 6$ ), confirming divisibility.
Identify whether $4,572$ is divisible by $3$ .

Yes, because $4+5+7+2=18$ and $18$ is divisible by $3$ . The sum 18 is divisible by 3 since $18 \div 3 = 6$ , applying the rule for 3.
Identify whether $5,376$ is divisible by $8$ .

Yes, because the last three digits $376$ are divisible by $8$ . Applying the rule for 8, 376 divided by 8 equals 47, an integer, confirming divisibility.
Identify whether $93,615$ is divisible by $5$ .

Yes, because the last digit is $5$ . The rule for 5 is satisfied as the number ends in 5, making it divisible by 5.
Identify whether $73,205$ is divisible by $2$ .

No, because the last digit $5$ is odd. The units digit 5 is odd, so the number is not even and thus not divisible by 2.
State the divisibility rule for $8$ .

Divisible by $8$ iff the last three digits form a multiple of $8$ . Since $1000 \equiv 0 \pmod{8}$ , divisibility by 8 is determined by the last three digits being a multiple of 8.
What is the remainder when $10^7$ is divided by $9$ ?

$1$ . Since $10 \equiv 1 \pmod{9}$ , $10^7 \equiv 1 \pmod{9}$ , yielding remainder 1.
State the divisibility rule for $4$ .

Divisible by $4$ iff the last two digits form a multiple of $4$ . Since $100 \equiv 0 \pmod{4}$ , divisibility by 4 depends only on the last two digits forming a multiple of 4.
State the divisibility rule for $11$ using alternating digit sums.

Divisible by $11$ iff the alternating sum of digits is a multiple of $11$ . Since $10 \equiv -1 \pmod{11}$ , the alternating sum of digits determines congruence modulo 11.
Identify whether $10^4+10^2+1$ is divisible by $3$ .

Yes, $10^4+10^2+1=10101$ and $1+0+1+0+1=3$ . The sum of digits is 3, which is divisible by 3 ( $3 \div 3 = 1$ ), confirming divisibility.
State the divisibility rule for $2$ .

Divisible by $2$ iff the last digit is even ( $0,2,4,6,8$ ). A number is divisible by 2 if it is even, which is determined solely by the parity of its units digit.
State the divisibility rule for $9$ .

Divisible by $9$ iff the sum of digits is divisible by $9$ . Since $10 \equiv 1 \pmod{9}$ , a number is congruent to the sum of its digits modulo 9.
Identify whether $10^5+10^3$ is divisible by $10$ .

Yes, $10^5+10^3=101000$ ends in $0$ . The expression simplifies to 101000, which ends in 0, satisfying the rule for 10.
Identify whether $2,418$ is divisible by $6$ .

Yes, it is divisible by $2$ and $3$ . It satisfies the rules for both 2 (even) and 3 (sum 15 divisible by 3), hence divisible by 6.
State the divisibility rule for $3$ .

Divisible by $3$ iff the sum of digits is divisible by $3$ . Since $10 \equiv 1 \pmod{3}$ , a number is congruent to the sum of its digits modulo 3.
Identify whether $47,190$ is divisible by $9$ .

No, because $4+7+1+9+0=21$ and $21$ is not divisible by $9$ . The sum 21 is not divisible by 9 (between 18 and 27), so the number is not divisible by 9.
Identify whether $7,128$ is divisible by $4$ .

Yes, because the last two digits $28$ are divisible by $4$ . Applying the rule for 4, 28 divided by 4 equals 7, an integer, confirming divisibility.
What is the remainder when $10^5+7$ is divided by $3$ ?

$2$ . Since $10 \equiv 1 \pmod{3}$ , $10^5 +7 \equiv 1+7=8 \equiv 2 \pmod{3}$ .

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