Question

let the natural number n have the decimal numeral 123,480,38d, where d is the units digit. use divisibility tests to give all of the choices of d by which n is divisible. complete parts (c) through (h) below.

(c) for what value(s) of d is n divisible by 4?
(use a comma to separate answers as needed.)

(d) for what value(s) of d is n divisible by 5?
(use a comma to separate answers as needed.)

(e) for what value(s) of d is n divisible by 6?
(use a comma to separate answers as needed.)

(f) for what value(s) of d is n divisible by 8?
(use a comma to separate answers as needed.)

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

Using the Divisibility Rules knowledge point
The natural number is \(n = 123,480,38d\), where \(d \in \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\).
We need to determine the values of \(d\) for which \(n\) is divisible by \(4\), \(5\), \(6\), \(9\), \(3\), and \(10\).

Divisibility by 4 and 5

Using the Divisibility Rules knowledge point

• For divisibility by \(4\), the last two digits \(8d\) must form a number divisible by \(4\).

Since \(80, 84, 88\) are divisible by \(4\), we have \(d \in \{0, 4, 8\}\).

• For divisibility by \(5\), the units digit \(d\) must be \(0\) or \(5\).

Thus, \(d \in \{0, 5\}\).

Divisibility by 6 and 9

Using the Divisibility Rules knowledge point

• For divisibility by \(9\), the sum of the digits must be divisible by \(9\).

Sum of digits: \(1 + 2 + 3 + 4 + 8 + 0 + 3 + 8 + d = 29 + d\).
For \(29 + d\) to be divisible by \(9\), we need \(29 + d = 36 \implies d = 7\).

• For divisibility by \(6\), \(n\) must be divisible by both \(2\) and \(3\).
• Divisibility by \(2\) requires \(d \in \{0, 2, 4, 6, 8\}\).
• Divisibility by \(3\) requires the sum of digits \(29 + d\) to be divisible by \(3\).

Thus, \(29 + d \in \{30, 33, 36\} \implies d \in \{1, 4, 7\}\).

• Intersecting these sets: \(d \in \{0, 2, 4, 6, 8\} \cap \{1, 4, 7\} = \{4\}\).

Divisibility by 3 and 10

Using the Divisibility Rules knowledge point

• For divisibility by \(3\), the sum of digits \(29 + d\) must be divisible by \(3\).

Thus, \(d \in \{1, 4, 7\}\).

• For divisibility by \(10\), the units digit \(d\) must be \(0\).

Thus, \(d = 0\).

Answer:

Part	Question	Answer
(d)	For what value(s) of \(d\) is \(n\) divisible by 5?	0, 5
(e)	For what value(s) of \(d\) is \(n\) divisible by 6?	4
(f)	For what value(s) of \(d\) is \(n\) divisible by 9?	7
(g)	For what value(s) of \(d\) is \(n\) divisible by 3?	1, 4, 7
(h)	For what value(s) of \(d\) is \(n\) divisible by 10?	0