Divisibility Tests

So far, we have been finding factors of numbers in different contexts, including to determine if a number is prime or not, or if a given pair of numbers is co-prime or not.

It is easy to find factors of small numbers. How do we find factors of a large number?

Let us take 8560. Does it have any factors from 2 to 10 (2, 3, 4, 5, ..., 9, 10)?

It is easy to check if some of these numbers are factors or not without doing long division. Can you find them? yesNo

Divisibility by 10

Let us take 10. Is 8560 divisible by 10 ? This is another way of asking if 10 is a factormultiple of 8560.

For this, we can look at the pattern in the multiples of 10.

The first few multiples of 10 are: 10, 20, 30, 40, … Continue this sequence and observe the pattern.

Is 125 a multiple of 10? Will this number appear in the previous sequence? Why or why not?

Answer: 125 is notis a multiple of 10 because it does not end with . Therefore, it will not appear in the sequence of multiples of 10.

Can you now answer if 8560 is divisible by 10?

Answer: YesNo, 8560 isis not divisible by 10 because it ends with .

Consider this statement: Numbers that are divisible by 10 are those that end with '0'. Do you agree?

Answer: yesNo,

I agree. A number is divisible by 10 if and only if its last digit is .

For example: 10, 120, 8560 are divisible not divisible by 10, while 125 and 43 are not.

Divisibility by 5

The number 5 is another number whose divisibility can easily be checked. How do we do it?

We check if a number is divisible by 5 by looking at its digit.

If the last digit is or , then the number is divisible by . Explore by listing down the multiples: 5, 10, , , 25,...

What do you observe about these numbers? Do you see a pattern in the last digit? YesNo, all multiples of 5 end with either or .

What is the largest number less than 399 that is divisible by 5? Is 8560 divisible by 5? The Largest Number is (since 395 ÷ 5 = ).

YesNo 8560 is divisible by 5 because it ends with .

Consider this statement:

Numbers that are divisible by 5 are those that end with either a '0' or a '5'.Do you agree? YesNo

I agree.
A number is divisible by 5 if and only if its last digit is or .

For example: 25, 40, 395, and 8560 are divisible by , while 123 and 48 are not.

Divisibility by 2

The first few multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ... What do you observe? Do you see a pattern in the last digit ?

Is 682 divisible by 2? Can we answer this without doing the long division?

Is 8560 divisible by 2 ? Why or why not ?

Consider this statement: Numbers that are divisible by 2 are those that end with '0', '2', '4', '6' or '8'. Do you agree?

What are all the multiples of 2 between and ?

Divisibility by 4

Checking if a number is divisible by can also be done easily!

Look at its multiples: 4, 8, 12, , , , 28, , …

Are you able to observe any patterns that can be used?

The multiples of 10, 5 and 2 have a pattern in their last digits which we are able to use to check for divisibility.

Similarly, can we check if a number is divisible by 4 by looking at the last digit?

It does not work! Look at 12 and 22. They have the same last digit, but 12 is a multiple of 4 while 22 is not. Similarly 14 and 24 have the same last digit, but 14 is not a multiple of 4 while 24 is. Similarly, 16 and 26 or 18 and 28. What this means is that by looking at the last digit, we cannot tell whether a number is a multiple of 4.

Can we answer the question by looking at more digits? Make a list of multiples of 4 between 1 and 200 and search for a pattern.

Divisibility by 8

Interestingly, even checking for divisibility by 8 can be simplified.

Can the last two digits be used for this?YesNo

I'll solve the original question about finding numbers divisible by 8 in the given ranges using the strict markdown syntax you've shown.

Find numbers between 120 and 140 that are divisible by 8. Also find numbers between 1120 and 1140, and 3120 and 3140, that are divisible by 8. What do you observe?

Answer:

Range 120-140: To find numbers divisible by 8, I'll check each number:

120 ÷ 8 = 15, 128 ÷ 8 = 16, 136 ÷ 8 = 17

Numbers divisible by 8: , ,

Range 1120-1140:

1120 ÷ 8 = 140, 1128 ÷ 8 = 141, 1136 ÷ 8 = 142

Numbers divisible by 8: , ,

Range 3120-3140:

3120 ÷ 8 = 390, 3128 ÷ 8 = 391, 3136 ÷ 8 = 392

Numbers divisible by 8: , ,

Observation:

Looking at the last two digits of each set:

• Range 120-140: 20, 28, 36

• Range 1120-1140: 20, 28, 36

• Range 3120-3140: 20, 28, 36

The pattern shows that the last two digits are identical in all three ranges: 20, 28, 36

This occurs because divisibility by 8 depends only on the last threetwo digits of a number. Since all ranges have the same pattern in their last two digits (×20 to ×40), the numbers divisiblenot divisible by 8 will have the same last two digits regardless of what comes before them.

🏵️Change the last two digits of 8560 so that the resulting number is a multiple of 8.

Answer:

Which two-digit numbers make 85__ divisible by 8:

8500 ÷ 8 = 1062 remainder 4

8501 ÷ 8 = remainder 5

8502 ÷ 8 = [1062] remainder 6

8503 ÷ 8 = remainder 7

8504 ÷ 8 = remainder 0

8512 ÷ 8 = remainder 0

8520 ÷ 8 = remainder 0

8528 ÷ 8 = remainder 0

8536 ÷ 8 = remainder 0

8544 ÷ 8 = remainder 0

8552 ÷ 8 = remainder 0 and soon..

The last two digits can be: , , , , , , , , , , ,

So 8560 can become: 8504, 8512, 8520, 8528, 8536, 8544, 8552, 8560, 8568, 8576, 8584, or 8592

Note: 8560 is already divisible by 8! (8560 ÷ 8 = )

1. 2024 is a leap year (as February has 29 days). Leap years occur in the years that are multiples of 4, except for those years that are evenly divisible by 100 but not 400.

a. From the year you were born till now, which years were leap years?

Answer: Since I don't know your birth year, here are examples:

If you were born in 2010: 2012, 2016, 2020,

If you were born in 2005: 2008, 2012, 2016, 2020,

b. From the year 2024 till 2099, how many leap years are there?

Answer: Leap years happen every 4 years. From 2024 to 2099: 2024, 2028, 2032, 2036, 2040, 2044, 2048, 2052, 2056, 2060, 2064, 2068, 2072, 2076, 2080, 2084, 2088, 2092, 2096

Total = leap years

2. Find the largest and smallest 4-digit numbers that are divisible by 4 and are also palindromes.

Answer:

A palindrome reads the same forwards and backwards (like 1221).

Smallest: Let me check 4-digit palindromes starting from 1001:

• 1001 ÷ 4 = 250.25 (has remainder)

• 1111 ÷ 4 = 277.75 (has remainder)

• 1221 ÷ 4 = 305.25 (has remainder)

• 2112 ÷ 4 = 528 (no remainder!)

Smallest =

Largest: Let me check from 9999 going down:

• 9999 ÷ 4 = 2499.75 (has remainder)

• 9889 ÷ 4 = 2472.25 (has remainder)

• 8888 ÷ 4 = 2222 (no remainder!)

Largest =

3. Explore and find out if each statement is always true, sometimes true or never true.

a. Sum of two even numbers gives a multiple of 4.

Answer: Sometimes true

Let me check with examples:

• 2 + 4 = 6 (6 ÷ 4 = 1 remainder 2)

• 4 + 8 = 12 (12 ÷ 4 = 3)

• 6 + 10 = 16 (16 ÷ 4 = 4)

b. Sum of two odd numbers gives a multiple of 4.

Answer:

Let me check with examples:

• 1 + 3 = 4 (4 ÷ 4 = 1)

• 3 + 5 = 8 (8 ÷ 4 = 2)

• 1 + 5 = 6 (6 ÷ 4 = 1 remainder 2)

4. Find the remainders when dividing by 10, 5, and 2.

Answer:

Tip: For ÷10, remainder is the last digit. For ÷2, remainder is 0 if even, 1 if odd.

5. Guna checked only two numbers to know 14560 is divisible by all of 2, 4, 5, 8 and 10. Which two?

Answer:

The two numbers are and .

Why this works:

• If a number divides by 8, it automatically divides by and
• If a number divides by 10, it automatically divides by
• So checking just 8 and 10 tells us about all five numbers!

6. Which numbers are divisible by ALL of 2, 4, 5, 8 and 10?

Answer:

572: Ends in 2, so not divisible by 5 → 2352: Ends in 2, so not divisible by 5 →
5600: Ends in 0 (✓ for 2,5,10), 600÷8=75 (✓ for 8), 00÷4=0 (✓ for 4) → 6000: Ends in 0 (✓ for 2,5,10), 000÷8=0 (✓ for 8), 00÷4=0 (✓ for 4) → 77622160: Ends in 0 (✓ for 2,5,10), 160÷8=20 (✓ for 8), 60÷4=15 (✓ for 4) →

7. Write two numbers whose product is 10000. Neither should end in 0.

Answer:

I need to find factors of 10000 that don't end in 0.

10000 = 10 × 10 × 10 × 10 = (2 × 5) × (2 × 5) × (2 × 5) × (2 × 5) 10000 = 2⁴ × 5⁴ = 16 × 625

Answer: and

Check: 16 × 625 = 10000 Neither 16 nor 625 ends in 0