Probability
7-digit numbers with digit sum 59 — divisibility by 11
MJAT_TS5_P2
Grade 12
Question:
The probability that a 7-digit number whose digit sum is 59 is divisible by 11 is $\dfrac{a}{b}$ where $\gcd(a,b)=1$. If $a+b=k$, then which is/are correct?
A) $k$ is the square of an integer
B) $k$ is a two-digit number
C) $k$ is a multiple of 10
D) $k$ is a composite number
Step-by-Step Solution
Key Concept: 7-digit numbers with digit sum 59: each digit is 1-9 (most must be large). Since max digit sum with 7 digits is 63, sum=59 means digits average ~8.4. Count such numbers total and divisible by 11 using generating functions.
$k$ is a 2-digit perfect square composite number. A ✓, B ✓, D ✓, C ✗. Answer: A, B, D.
Correct Answer: ABD