Question

let the natural number n have the decimal numeral 123,454,26d, where d is the units digit. use divisibility tests to give a parts (a) through (h) below.

0,4,8 (use a comma to separate answers as needed.)

(d) for what value(s) of d is n divisible by 5?

0,5 (use a comma to separate answers as needed.)

(e) for what value(s) of d is n divisible by 6?

0,6 (use a comma to separate answers as needed.)

(f) for what value(s) of d is n divisible by 8?

4 (use a comma to separate answers as needed.)

(g) for what value(s) of d is n divisible by 9?

0,9 (use a comma to separate answers as needed.)

(h) for what value(s) of d is n divisible by 10?

(use a comma to separate answers as needed.)

Explanation:

Analyze the number and divisibility by 5

The natural number is \(n = 123,454,26d\).
Using the Divisibility Rules concept, a number is divisible by 5 if and only if its units digit \(d\) is either 0 or 5.
Thus, the values of \(d\) for divisibility by 5 are 0, 5.

Analyze divisibility by 6

Using the Divisibility Rules concept, a number is divisible by 6 if it is divisible by both 2 and 3.
For divisibility by 2, \(d\) must be even: \(d \in \{0, 2, 4, 6, 8\}\).
For divisibility by 3, the sum of the digits must be divisible by 3.
The sum of the digits is:

$$ 1 + 2 + 3 + 4 + 5 + 4 + 2 + 6 + d = 27 + d $$

Since 27 is divisible by 3, \(27 + d\) is divisible by 3 if \(d\) is divisible by 3: \(d \in \{0, 3, 6, 9\}\).
Intersecting the two sets:

$$ d \in \{0, 2, 4, 6, 8\} \cap \{0, 3, 6, 9\} = \{0, 6\} $$

Thus, the values of \(d\) for divisibility by 6 are 0, 6.

Analyze divisibility by 8

Using the Divisibility Rules concept, a number is divisible by 8 if the number formed by its last three digits is divisible by 8.
The last three digits form the number \(26d\), which is \(260 + d\).
We divide 260 by 8:

$$ 260 = 8 \times 32 + 4 $$

So, \(26d = 8 \times 32 + (4 + d)\).
For \(26d\) to be divisible by 8, \(4 + d\) must be divisible by 8.
Since \(d\) is a single digit (\(0 \le d \le 9\)):

$$ 4 + d = 8 \implies d = 4 $$

Thus, the value of \(d\) for divisibility by 8 is 4.

Analyze divisibility by 9

Using the Divisibility Rules concept, a number is divisible by 9 if the sum of its digits is divisible by 9.
The sum of the digits is \(27 + d\).
Since 27 is divisible by 9, \(27 + d\) is divisible by 9 if \(d\) is divisible by 9.
Since \(d\) is a single digit, \(d\) can be 0 or 9.
Thus, the values of \(d\) for divisibility by 9 are 0, 9.

Analyze divisibility by 10

Using the Divisibility Rules concept, a number is divisible by 10 if and only if its units digit \(d\) is 0.
Thus, the value of \(d\) for divisibility by 10 is 0.

Answer:

Question d

0,5

Question e

0,6

Question f

4

Question g

0,9

Question h

0