Question

(a) at a glance, determine the digit d so that 63,416,12d is divisible by 4. is there more than one answer?
(b) can you choose the digit d so that 63,416,12d is divisible by 8 and not 16?

(a) which of the following lists all of the possible values for the digit d?
a. the digit d can be 0, 2, 4, 6, or 8.
b. the digit d can be 2.
c. the digit d can be 0.
d. the digit d can be 6.
e. the digit d can be 8.
f. the digit d can be 0, 2, or 6.
g. the digit d can be 4.
h. the digit d can be 0, 4, or 8.
i. the digit d can be 2 or 6.
j. the digit d can be 4 or 8.

(b) is there a digit d such that the number is divisible by 8 and not 16?
a. yes, if the digit d is 0.
b. yes, if the digit d is 4.
c. no
d. yes, if the digit d is 6.
e. yes, if the digit d is 2.
f. yes, if the digit d is 8.

Explanation:

Apply divisibility rule for 4

Using the Divisibility Rules knowledge point

$$
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$$

Analyze divisibility by 8 and 16

We test the candidate digits \(d \in \{0, 4, 8\}\) for the number \(N = 63,416,12d\).
A number is divisible by 8 if its last 3 digits form a number divisible by 8.
A number is divisible by 16 if its last 4 digits form a number divisible by 16.

For \(d = 0\):

• Last 3 digits: \(120\). Since \(120 = 8 \times 15\), \(N\) is divisible by 8.
• Last 4 digits: \(6120\). Since \(6120 = 16 \times 382.5\), \(N\) is not divisible by 16.

Thus, \(d = 0\) satisfies the condition.

For \(d = 4\):

• Last 3 digits: \(124\). Since \(124 = 8 \times 15.5\), \(N\) is not divisible by 8.

For \(d = 8\):

• Last 3 digits: \(128\). Since \(128 = 8 \times 16\), \(N\) is divisible by 8.
• Last 4 digits: \(6128\). Since \(6128 = 16 \times 383\), \(N\) is divisible by 16.

Thus, \(d = 8\) does not satisfy the "not 16" condition.

Match options for both questions

• For part (a), the possible values for \(d\) are 0, 4, or 8. This matches option H.
• For part (b), the digit \(d\) must be 0. This matches option A.

Answer:

Question 1

• A. The digit d can be 0, 2, 4, 6, or 8.
• B. The digit d can be 2.
• C. The digit d can be 0.
• D. The digit d can be 6.
• E. The digit d can be 8.
• F. The digit d can be 0, 2, or 6.
• G. The digit d can be 4.
• H. The digit d can be 0, 4, or 8. (Correct answer)
• I. The digit d can be 2 or 6.
• J. The digit d can be 4 or 8.

Question 2

• A. Yes, if the digit d is 0. (Correct answer)
• B. Yes, if the digit d is 4.
• C. No.
• D. Yes, if the digit d is 6.
• E. Yes, if the digit d is 2.
• F. Yes, if the digit d is 8.