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Problems on Divisibility Rules
Problems on divisibility rules will help us to learn how to use the rules to test of divisibility by 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11.
1. Is 7248 is divisible (i) by 4, (ii) by 2 and (iii) by 8?
(i) The number 7248 has 48 on its extreme right side which is exactly divisible by 4. When we divide 48 by 4 we get 12.
Therefore, 7248 is divisible by 4.
(ii) The number 7248 has 8 on its unit place which is an even number so, 7248 is divisible by 2.
(iii) 7248 is divisible by 8 as 7248 has 248 at its hundred place, tens place and unit place which is exactly divisible by 8.
2. A number is divisible by 4 and 12. Is it necessary that it will be divisible by 48? Give another example in support of you answer.
48 = 4 × 12 but 4 and 12 are not co-prime.
Therefore, it is not necessary that the number will be divisible by 48.
Let us consider the number 72 for an example
72 ÷ 4 = 18, so 72 is divisible by 4.
72 ÷ 12 = 6, so 72 is divisible by 12.
But 72 is not divisible by 48.
3. Without actual division, find if 235932 is divisible (i) by 4 and (ii) 8.
(i) The number formed by the last two digits on the extreme right side of 235932 is 32.
32 ÷ 4 = 8, i.e. 32 is divisible by 4.
Therefore, 235932 is divisible by 4.
(ii) The number formed by the last three digits on the extreme right side of 235932 is 932.
But 932 is not divisible by 8.
Therefore, 235932 is not divisible by 8.
4. Test whether the following numbers are divisible by 3 or 9:
(i) 546
(ii) 8253
(iii) 43399
Solution:
(i) The given number is 546.
The sum of digits 5 + 4 + 6 = 15.
Since 15 is divisible by 3, so 546 is divisible by 3.
Since 15 is not divisible by 9, so 546 is not divisible by 9.
(ii) The given number is 8253.
The sum of digits 8 + 2 + 5 + 3 = 18.
Since 18 is divisible by both 3 and 9, therefore 8253 is divisible by 3 and 9.
(iii) The given number is 43399.
The sum of digits = 4 + 3 + 3 + 9 + 9 = 28.
Since 28 is not divisible by 3 and 9, therefore 43399 is neither divisible by 3 nor by 9.
5. Which of the following numbers are divisible by 5 and 10?
(i) 3527
(ii) 13520
(iii) 37595
Solution:
(i) In 3527, the digit at ones place is neither 0 nor 5.
Therefore, 3527 is not divisible by 5 and 10.
(ii) In 13520, the digit at ones place is 0.
Therefore, 13520 is divisible by 5 and 10.
(iii) In 37595, the digit at ones place is 5.
Therefore, 37595 is divisible by 5.
But the digit at ones place is not 0, so 37595 is not divisible by 10.
6. Find whether or not 3414 is divisible by:
(i) 2
(ii) 3
(iii) 5
(iv) 9.
Solution:
(i) In 3414, the digit at ones place is 4.
Therefore, 3414 is divisible by 2.
(ii) The sum of digits = 3 + 4 + 1 + 4 = 12, which is a multiple of 3.
Therefore, 3414 is divisible by 3.
(iii) In 3414, the digit at ones place is neither 0 nor 5.
Therefore, 3414 is not divisible by 5.
(iv) Since the sum of digits = 3 + 4 + 1 + 4 = 12, which is not divisible by 9.
Therefore, 3414 is not divisible by 9.
7. Find the largest prime number that you need to test as a divisor to find whether or not 211 is a prime number.
Solution:
The prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, etc.
Let us test the divisibility of 211 by these primes:
211 is not divisible by 2, because it is not an even number.
211 is not divisible by 3, as 2 + 1 + 1 = 4 is not divisible by 3.
211 is not divisible by 5, as the last digit is not 0 or 5.
211 is not divisible by 7, as 211 ÷ 7 = 30, leaving remainder 1.
211 is not divisible by 11, as 211 ÷ 11 = 19, leaving remainder 2.
211 is not divisible by 13, as 211 ÷ 13 = 16, leaving remainder 3.
The next prime number is 17, which is greater than the last quotient 16.
So we need not test its divisibility by 17.
Thus, the greatest prime number required to test whether 211 is prime or not is 13.
Since 211 is not divisible by 13, so it is a prime number.
8. Fill in the blanks with the smallest number to make the following number divisible by 9:
78 _ 964.
Solution:
We have 78 _ 964
The sum of digits = 7 + 8 + 9 + 6 + 4 = 34
The next number nearest to 34 which is divisible by 9 is 36.
Difference = 36 - 34 = 2, which is the required number.
Hence, the number which is divisible by 9 is 782964.
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