Numbers and Divisibility

A number is divisible by 7 if unit's place digit is multiplied by 2 and subtracted from the remaining digits and the number obtained is divisible by 7. 47331-   (2*2) = 473274732 - (7*2) = 4718471 - (8*2) = 45545 - (5*2) = 3535 is divisible by 7.Therefore, 473312 is divisible by 7. It is the correct answer.

It is not divisible by 2.

Numbers between 500 and 1000 divisible by 13 are: 507 (13×39), 520 (13×40), ..., 988 (13×76). This is an AP with first term 507, last term 988, and 38 terms. Sum = 38/2 × (507 + 988) = 19 × 1495 = 28405.

Use inclusion-exclusion: Sum of multiples of 4 (10200) + Sum of multiples of 5 (8200) - Sum of multiples of 20 (2200, counted twice). Total = 10200 + 8200 - 2200 = 16200.

Numbers divisible by both 4 and 5 are divisible by LCM(4,5)=20. Multiples of 20 from 100 to 300 are: 100, 120, ..., 300. This is 11 terms. Sum = 11/2 × (100 + 300) = 11/2 × 400 = 2200.

x + 2 is a factor             x = - 2 makes it zero

(-2)³ + 6(-2)² + 4(-2) + k = 0       i.e k = - 8

To find greatest which is divisible by 21, 18, 12 and 15, we shall have to find the LCM of 21-18-12-15 which is 1260. After dividing 9999 by 1260 we get 1179 remainder. So answer will be 9999-1179 = 8820

n³ - n = (n - 1) (n) (n + 1), for integral values of n, represents the product of three consecutive integers. Since in a pair of consecutive integers, there is a multiple of two and in a triplet of consecutive integers, there is a multiple of three, then n³ - n is divisible by 6. Hence, option (5) is the correct option.

Options 2 and 4 both are correct as for m = 1 and 2, the numbers so formed are not divisible by 3.

214984 + 2 = 214986 (divisible by 3)

13 K + 1339 L = 13 (K + 103)

If a number is divisible by both 2 and 3 then it is divisible by 6 also

On applying the divisibility rule of 11, We get, sum of odd terms=8 and sum of even terms=8 Difference= 8-8=0, which is divisible by 11

[Explanation: Find the sum of digits as 6+3+9+2+1+0=21 which divisible by 3 but not by 9]

[Explanation: On applying the divisibility rule of 8, i.e., last 3 digits of the number are divided By 8, 176/8= 22 and remainder as 0 This means, 37450176 which is divisible by 8]