A number is exactly divisible by both 11 and 13. Identify which statement must be true about this number.

Introduction / Context:This is a fundamental divisibility question about combined factors. When a number is divisible by two integers, we investigate what stronger conditions follow, particularly relating to their product and least common multiple (LCM).

Given Data / Assumptions:

• The number is divisible by 11.
• The same number is divisible by 13.
• 11 and 13 are distinct primes.

Concept / Approach:If a number N is divisible by prime p and prime q, then N is divisible by p*q, because primes have no nontrivial common factors. In general, N is divisible by LCM(11, 13). With primes, LCM equals their product 11*13 = 143.

Step-by-Step Solution:Since 11 | N and 13 | N, and gcd(11, 13) = 1, LCM(11, 13) = 11*13.Therefore, N is divisible by 11*13 = 143.

Verification / Alternative check:Take N = 143 as the smallest example. It is divisible by both 11 and 13. Any integer multiple of 143 also shares this property, confirming the rule.

Why Other Options Are Wrong:Divisible by (11 + 13) = 24 is not guaranteed. Divisible by (13 - 11) = 2 is not guaranteed. Divisible by (13 ÷ 11) is meaningless for integers. “Divisible by 22 only” ignores the 13 condition.

Common Pitfalls:Confusing sum or difference with necessary divisibility; forgetting that for primes, the LCM equals the product; and misreading divisibility statements involving division.

Final Answer:Divisible by (11 x 13)