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The difference between two sums = 20 + * – 15 = 5 + *, which should be divisible by 11.

So, 5 + * = 11

Thus, the required digit is 6.
Example 17: Find a 4-digit odd number using each of the digits 1, 2, 4 and 5 only once such that
when the first and the last digits are interchanged, it is divisible by 4.

Solution: For a 4-digit odd number using the digits 1, 2, 4 and 5, we can put 1 or 5 at ones place
and other three digits at higher places in any way.
But, the number formed after interchanging the thousands and ones place should be divisible by
4. So, we can put only 2 or 4 at thousands place.

Therefore, we have 2145, 2415, 2451, 2541,
4125, 4215, 4251, 4521 Think and Answer

After interchanging the first and the last Using each of the digits 1, 2, 3 and 4 only once,
digits, we have 5142, 5412, 1452, 1542, 5124, determine the smallest 4-digit number divisible by 4.
5214, 1254, 1524
Among these numbers, only 5412, 1452, 5124, and 1524 are divisible by 4.

Hence, anyone out of these four numbers: 2415, 2451, 4125, and 4521 can be taken.
Using Divisibility Rules to Determine a Prime Number

To determine a number greater than 100 whether it is a prime number or a composite number,
we divide the number by the consecutive prime numbers, till we get quotient less than divisor.
We can also use the divisibility rules. Let us understand it with the help of examples.
Example 18: Identify whether the given numbers are prime or composite.

(a) 139 (b) 161 (c) 289
Solution: Divide the number by the prime numbers 2, 3, 5, 7, 11, 13 and so on.

(a) 139 is an odd number. So, it is not divisible by 2.

1 + 3 + 9 = 13, so it is not divisible by 3.
Further, its ones digit is 9 so it is not divisible by 5.

By long division for 7, 19
7 139
139 ÷ 7 leaves remainder 6. So it is not divisible by 7. – 7
Next, (1 + 9) – 3 = 7, which is not divisible by 11. So, 139 is not divisible – 69
by 11. 63
6
By long division for 13,
10
139 ÷ 13 leaves remainder 9. So it is not divisible by 13. 13 139
Here, quotient 10 < divisor 13. So we will not try ahead. – 13
09
Thus, 139 is a prime number. – 00
(b) 161 is divisible by 7. Therefore, 161 is a composite number. 9

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