Why the units digit decides divisibility by 10
Every place value but the units is a multiple of 10.
Which digit matters?
Take the number 4075, written dcba. Its place values are 1000, 100, 10 and 1. To test whether it is divisible by 10, we divide it into groups of 10.
Which digit of 4075 decides if it is divisible by 10?
What this lesson covers
The idea
Every place value except the units is a multiple of 10, so a number is divisible by 10 if and only if its units digit is 0; for 2, 5 and 10 the units digit alone decides divisibility.
Which digit matters?
Take the number 4075, written dcba. Its place values are 1000, 100, 10 and 1. To test whether it is divisible by 10, we divide it into groups of 10.
Which digit of 4075 decides if it is divisible by 10?
- The units digit, 5
- The thousands digit, 4
- All four digits together
Split off the units
The number is split into tens and above and the units digit a. Pick a divisor: 10, 5 or 2. Change digits with ▲ ▼ and watch the remainders.
Only the units can leave a remainder
Every place value except the units is a multiple of 10, so a number is divisible by 10 if and only if its units digit is 0; for 2, 5 and 10 the units digit alone decides divisibility.
In dcba, the part 1000d + 100c + 10b is always a multiple of 10. It is also a multiple of 5 and of 2, since 5 and 2 divide 10. Whatever is left over comes only from a.
So: divisible by 10 when a = 0 divisible by 5 when a = 0 or a = 5 divisible by 2 when a is even
Notes
Every place value except the units is a multiple of 10, so a number is divisible by 10 if and only if its units digit is 0; for 2, 5 and 10 the units digit alone decides divisibility.
Check yourself
The number dcba has a = 3. Is it divisible by 10, whatever d, c, b are?
How many of 1250, 3475, 6082, 9910 are divisible by 5?
Answer: 3
The units digits are 0, 5, 2 and 0. A number is divisible by 5 when its units digit is 0 or 5, so 1250, 3475 and 9910 are. 6082 is not. So 3 numbers.
A number dcba is divisible by 2 and leaves remainder 1 when divided by 5. What is the units digit a?
Answer: 6
Remainder 1 on division by 5 means a is 1 or 6. Divisible by 2 means a is even. Only 6 is both.
Why can the hundreds digit c never change whether a number is divisible by 10?
- Yes, if d is even. The digit d does not matter. 1000d is a multiple of 10 for every d.
- No, never — correct. Yes! 1000d + 100c + 10b is a multiple of 10 and 3 is left over, so the remainder is 3 and the number is never divisible by 10.
- Yes, if b = 0. Even with b = 0, the 3 ones are still left over. Only a = 0 makes a number divisible by 10.
- Because c is always 0. c can be any digit from 0 to 9.
- Because 100c is a multiple of 10 for every c, so it never leaves anything over — correct. Yes! 100c = 10 × 10c is always made of whole groups of 10. Only the units digit can leave a remainder.
- Because the hundreds digit is small. Size does not matter. 100c is large, but it is always a multiple of 10.