November 2025 Contest — Grade 8

Question 1

A digital clock shows the time using four digits (for example, 09:45).
Every time the minute increases by 1, some of the digits on the display change. Between 12:00pm and 1:00pm, how many times does exactly one digit change when the time goes up by one minute?

Accepted Answer

To find how many times exactly one digit changes on the digital clock between 12:00pm and 1:00pm, we can analyze the transitions minute by minute.

There are 60 minute-to-minute transitions in total during this hour, starting from 12:00 to 12:01, and ending with 12:59 to 01:00 (since the clock uses four digits, 1:00pm is displayed as 01:00).

Let's look at how many digits change in each transition:

1. Within each 10-minute block (e.g., from 12:00 to 12:09):
From 12:00 to 12:01, only the last digit changes (0 to 1). This is exactly 1 digit changing.
This 1-digit change happens for 12:00 to 12:01, 12:01 to 12:02, ..., up to 12:08 to 12:09.
This gives 9 transitions where exactly 1 digit changes.

2. At the end of each 10-minute block (except the last one):
From 12:09 to 12:10, the last two digits change (09 to 10). This is 2 digits changing.
Similarly, 2 digits change at 12:19 to 12:20, 12:29 to 12:30, 12:39 to 12:40, and 12:49 to 12:50.

3. At the end of the hour:
From 12:59 to 01:00, all four digits change (1 to 0, 2 to 1, 5 to 0, 9 to 0). This is 4 digits changing.

Now, we can count the total number of transitions where exactly 1 digit changes:
There are 6 blocks of 10 minutes: 
12:00 to 12:10
12:10 to 12:20
12:20 to 12:30
12:30 to 12:40
12:40 to 12:50
12:50 to 01:00
In each of these 6 blocks, exactly 9 transitions involve only 1 digit changing (the transitions ending in 1, 2, 3, 4, 5, 6, 7, 8, 9).

Thus, the total number of times exactly one digit changes is:
6 x 9 = 54

Alternatively, out of the 60 total transitions, there are 6 transitions where more than 1 digit changes (at the end of each 10-minute interval: 12:09, 12:19, 12:29, 12:39, 12:49, and 12:59). Therefore, the number of transitions with exactly 1 digit changing is:
60 - 6 = 54

The answer is B.

Question 2

The sum of the factors of a number is 39. Which of the following is true about the number?

Accepted Answer

To find the number whose sum of factors is 39, let us denote the number as N and the sum of its factors as sigma(N) = 39.

We know that the sum of factors function sigma(N) is multiplicative. If N is prime, then sigma(N) = N + 1 = 39, which gives N = 38. However, 38 is not a prime number (since 38 = 2 x 19), so N cannot be prime.

Let's express sigma(N) using the prime factorization of N. If N = p^a q^b dots, then:
sigma(N) = sigma(p^a) sigma(q^b) dots = 39

Since 39 can only be factored into integers greater than 1 as 3 x 13, we can set up the system:
1) sigma(p^a) = 3
2) sigma(q^b) = 13

Let's solve each part:
For sigma(p^a) = 1 + p + p^2 + dots + p^a = 3, the only integer solution is p = 2 and a = 1 (since 1 + 2 = 3).
For sigma(q^b) = 1 + q + q^2 + dots + q^b = 13, we can test values of b:
If b = 1, then 1 + q = 13 implies q = 12 (not prime).
If b = 2, then 1 + q + q^2 = 13 implies q(q + 1) = 12. Since 3 x 4 = 12, we find q = 3, which is a prime number.

Thus, the prime factorization of N is:
N = 2^1 x 3^2 = 2 x 9 = 18

Let's verify the factors of 18:
The factors of 18 are 1, 2, 3, 6, 9, and 18.
Their sum is 1 + 2 + 3 + 6 + 9 + 18 = 39.

Since the number is 18, it lies between 16 and 20.

The answer is B

November 2025 Contest — Grade 8

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