Which of the following numbers are divisible by all of 2, 4, 5, 8 and 10: 572, 2352, 5600, 6000, 77622160.

Question

Homi · Accepted Answer

We need to find numbers that can be divided by 2, 4, 5, 8, and 10 without any leftover.

Step 1 — Knowing the rules

Let us remember the rules for dividing numbers.

Divisibility by 2: A number can be divided by 2 if its last digit is 0, 2, 4, 6, or 8. These are even numbers.
Divisibility by 4: A number can be divided by 4 if the number made by its last two digits can be divided by 4.
Divisibility by 5: A number can be divided by 5 if its last digit is 0 or 5.
Divisibility by 8: A number can be divided by 8 if the number made by its last three digits can be divided by 8.
Divisibility by 10: A number can be divided by 10 if its last digit is 0.

Step 2 — Finding a common rule

A number must follow all these rules. Let us look at the rules for 5 and 10. For a number to be divisible by 10, its last digit must be 0. If its last digit is 0, it is also divisible by 5 (because it ends in 0). Also, if its last digit is 0, it is an even number, so it is divisible by 2. So, any number that can be divided by 2, 5, and 10 must end in 0.

Now we only need to check for divisibility by 4 and 8 for numbers that end in 0.

For 4: The last two digits must make a number divisible by 4.
For 8: The last three digits must make a number divisible by 8.

So, our main check is:

The number must end in 0.
The number formed by its last two digits must be divisible by 4.
The number formed by its last three digits must be divisible by 8.

Step 3 — Checking each number

Let us check each number from the list.

For 572: The last digit is 2. It does not end in 0. So, it is not divisible by 5 or 10. This number is not divisible by all.

For 2352: The last digit is 2. It does not end in 0. So, it is not divisible by 5 or 10. This number is not divisible by all.

For 5600:

The last digit is 0. (Divisible by 2, 5, 10)
The last two digits are 00. $00 \div 4 = 0$ 00 is divisible by 4. (Divisible by 4)
The last three digits are 600. $600 \div 8 = 75$ 600 is divisible by 8. (Divisible by 8) All conditions are met.

$\boxed{\text{5600 is divisible by all}}$

For 6000:

The last digit is 0. (Divisible by 2, 5, 10)
The last two digits are 00. $00 \div 4 = 0$ 00 is divisible by 4. (Divisible by 4)
The last three digits are 000. $000 \div 8 = 0$ 000 is divisible by 8. (Divisible by 8) All conditions are met.

$\boxed{\text{6000 is divisible by all}}$

For 77622160:

The last digit is 0. (Divisible by 2, 5, 10)
The last two digits are 60. $60 \div 4 = 15$ 60 is divisible by 4. (Divisible by 4)
The last three digits are 160. $160 \div 8 = 20$ 160 is divisible by 8. (Divisible by 8) All conditions are met.

$\boxed{\text{77622160 is divisible by all}}$

Answer

The numbers divisible by all of 2, 4, 5, 8, and 10 are:

5600 6000 77622160

Question 41

Answer

More questions in FIO