Question

In 154 N38 what digit should be replaced to N to make the number divisible by 3,6 and 9?​

Answer 1

Answer:

$N = 6$ so that number becomes divisible by 3, 6 and 9.

Step-by-step explanation:

In Number Theory there is a rule of thumb which states that sum of digits of a multiple of 3 equal 3 or a multiple of three. If we know that $n = 154N38$, then its sum of digits is:

$x = 1 + 5+4+N+3+8$

$x = 21+N$ (Eq. 1)

We have to determine which digits corresponds to multiples of three, there are four digits:

N = 0

$x = 21+0$

$x = 21$ ($3\times 7 = 21$)

N = 3

$x = 21+3$

$x = 24$ ($3\times 8 = 24$)

N = 6

$x = 21+6$

$x = 27$ ($3\times 9 = 27$)

N = 9

$x = 21+9$

$x = 30$ ($3\times 10 = 30$)

We get the following four distinct options: 154038, 154338, 154638, 154938. Now we find which number is divisible by 6 and 9 by factor decomposition:

$154038 = 2\times 3\times 25673$

$154338 =2\times 3\times 29 \times 887$

$154638 = 2\times 3\times 3\times 11\times 11\times 71$

$154938 = 2\times 3\times 7\times 7\times 17\times 31$

It is quite evident that $N = 6$ so that number becomes divisible by 3, 6 and 9.