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Divisibility
What happens when we divide 44 by 11 ? 44 ÷ 11 = _________
What about 126 ÷ 6 ? ______________
There is no remainder in either case. We say 44 is divisible by 11 and 126 is divisible by 6.
Divisibility by 2
When is a counting number divisible by 2 ?
Which of these are divisible by 2 ;
246
3467
122
3330
2191
Divisibility by 3:
To determine if a counting is divisible by 3: example 43259
1) add the digits : 4 + 3 + 2 + 5 + 9= 23
2) Repeat addition with the new result until you have a single digit: 2 + 3 = 5
( you can divide by 3 at any stage)
3) Divide the sum by 3: 5 ÷ 3
does not divide evenly but it does give you the remainder → 2
So the remainder of 43259 ÷ 3 is 2
What is the remainder when each of the following numbers is divided by 3
a) 94 ? __________
b) 142 ? __________
c) 3291 ? __________
HINT: You can discard part of the sum (digits) if they are already divisible by 3 )
94: since 9 is divisible by 3, just look at 4. 4 ÷ 3 has a remainder of ?
Divisibility by 4
The last two digits (right-most) have to be divisible by 4 in order for the given number to be divisible by 4
To find the remainder, divide these two digits by four. You can get the remainder from this division.
Is 448 divisible by 4 ?
yes, 48 ÷ 4 = __________ with no remainder
Is 9321 divisible by 4 ? no, 21 ÷ 4 has a remainder of ______ which means 9321 ÷ 4 has the same remainder
What is the reminder of
4 17 ÷ 4 ? _________
9469 ÷ 4 ? ___________
Divisibility by 5
If the number ends in a 5 or a zero, then the number is divisible by 5.
325, 4320, 1090, 45,... all of these numbers are divisible by 5
but
219 , 3 45 6,
123456 , .... are not
Use the last digit to find the remainder.
If the digit is less than 5, that’s the remainder → 341 ÷ 5 has a remainder of ________
4923 ÷ 5 has a remainder of ___________
If the digit is greater than 5, subtract 5 from that digit and that’s the remainder
359 ÷ 5 has a remainder of 9 – 5 = ________
9328 ÷ 5 has a remainder of 8 – 5 = _______
Divisibility by 8 :
Use only the last three digits ( rightmost digits )
Divide those digits by 8 – you can obtain the remainder
ex. 4342 ÷ 8 →
the quotient of 342 ÷ 8 is 42 with a remainder of 6, so the remainder of 4342 ÷ 8 is also 6
ex. What is the remainder of 5017 ÷ 8 ? → _____________
ex. Is 8 a divisor (factor ) of 2 168 ?
Is the remainder of 2168 ÷ 8 zero ? → ___________________
Divisibility by 9
Add all of the digits
If the result is not a single digit, then add all the digits . Continue until result is a single digit.
The single digit result represents the remainder. If the single digit is 9, then the remainder is zero.
ex.
What is the remainder of 92 ÷ 9 ? 9 + 2 = 11 → 11 is not a single digit →
continue 1 + 1 = 2 this represents the remainder.
ex. Find the remainder of 329 ÷ 9. 3 + 2 + 9 =14 → 14 is not a single digit → 1 + 4 = 5 → the remainder is 5
( Hint: when a 9 appears you can discard it without changing your answer )
Divisibility by 11 ( Study this method last – since it is not an obvious approach )
1) Add every other digit beginning with the one’s digit ( one’s, hundred’s, ten-thousands,... )
2) Add the remaining digits ( ten’s thousand’s, ... )
3) If the first sum is less than the second one, add 11 to the first sum
4) Subtract the first sum minus the second sum
The result we get here will represent the remainder.
Find the remainder of 24581 ÷ 11. 1 + 5 + 2 = 8 Also, 8 + 4 = 12 8 < 12, so add 11 to 8. Now 19 – 12 = 7
That’s the remainder.
The remainder of 3294 ÷ 11 is → 4 + 2 = 6 ; 9 + 3 = 12, ( 6 + 11 ) – 12 = 5 → the remainder = 5