Checking Divisibility by 2

Is 1,458 divisible by 2?

Look at the last digit: The last digit of $1{,}458$ is $8$ = Last digit: $8$

Check if it's even: Is $8$ in the set $\{0, 2, 4, 6, 8\}$? = Yes, $8$ is even

Apply the rule: Since the last digit is even, $1{,}458$ is divisible by $2$ = $1{,}458 \div 2 = 729$

Answer: Yes, 1,458 is divisible by 2 because its last digit (8) is even.

Checking Divisibility by 5

Which of these numbers are divisible by 5: 235, 482, 790?

Check 235: Last digit is $5$. Is $5$ equal to $0$ or $5$? = Yes! $235 \div 5 = 47$

Check 482: Last digit is $2$. Is $2$ equal to $0$ or $5$? = No. $482$ is NOT divisible by $5$

Check 790: Last digit is $0$. Is $0$ equal to $0$ or $5$? = Yes! $790 \div 5 = 158$

Answer: 235 and 790 are divisible by 5. 482 is not.

Divisibility by 10 and Patterns

A warehouse has 3,540 items. Can they be packed into boxes of 10 with none left over?

Identify what we need: We need to know if $3{,}540 \div 10$ has no remainder = Check divisibility by $10$

Apply the rule: Last digit of $3{,}540$ is $0$ = Last digit is $0$

Draw conclusion: Numbers ending in $0$ are always divisible by $10$ = $3{,}540 \div 10 = 354$ boxes

Answer: Yes! 3,540 items can be packed into exactly 354 boxes of 10.

Mistake: Looking at the first digit instead of the last digit

Why: For divisibility by 2, 5, and 10, only the last digit matters. The first digit tells us the size of the number, not its divisibility.

Correct: Always look at the rightmost (last) digit: in 847, check 7, not 8.

Mistake: Thinking a number ending in 5 is divisible by 10

Why: Divisibility by 10 is stricter than by 5. A number must end in 0 (not just 0 or 5) to be divisible by 10.

Correct: 75 is divisible by 5 but NOT by 10. Only 70, 80, 90... are divisible by 10.

Mistake: Confusing 'even' with 'divisible by 10'

Why: All numbers divisible by 10 are even, but not all even numbers are divisible by 10.

Correct: 24 is even (divisible by 2) but not divisible by 10. 20 is both even AND divisible by 10.

Sharing Equally

When splitting items between people, divisibility rules help you know instantly if it works.

Can 36 candies be shared equally between 2 children? Yes! $36$ ends in $6$ (even), so $36 \div 2 = 18$ each.

Counting Money

Money often uses denominations of 10, making divisibility by 10 very practical.

If you have 240 cents, can you exchange them for dimes (10 cents)? Yes! $240$ ends in $0$, so $240 \div 10 = 24$ dimes.

Q: Why does the last digit rule work?

A: Any number can be written as a multiple of 10 plus its last digit. For example, $347 = 340 + 7$. Since $340$ is always divisible by 2, 5, and 10, we only need to check if the last digit ($7$) is divisible.

Q: If a number is divisible by 10, is it always divisible by 2 and 5?

A: Yes! Since $10 = 2 \times 5$, any number divisible by 10 must also be divisible by both 2 and 5. For example, 60 is divisible by 10, 5, and 2.

Q: What about large numbers like 1,000,000?

A: The same rules apply! 1,000,000 ends in 0, so it's divisible by 2, 5, AND 10. These rules work for numbers of any size.