Question

Let a and b be integers, not both 0, and let t be the positive integer. Prove that tis the least common multiple of aand bif and only if tsatisfies these conditions:

1. $a|t$and $b|t$;
2. If $a|c$and $b|c$, then $t|c$.

Short answer

It is proved that is the least common multiple of and if and only if satisfies the following conditions:

1. $a|t$and $b|t$.
2. If $a|c$and $b|c$, $t|c$.

Step-by-step solution

Prove that it satisfies both conditions

It is given that t is the least common multiple of a and b, that is, $\left[a,b\right]=t$. According to the definition of the least common multiple, t is the multiple of a and b, that is, $t|a$and $t|b$.

When $a|c$and $b|c$, c is also a common multiple of a and b. This implies that it is a multiple of t. Therefore, $t|c$.

Hence, both conditions are satisfied.

Prove that if t|c when a|cand b|c, t is the least common multiple of a and b 

According to the given situation, both conditions are satisfied.

From the first condition, we have $a|t$and $b|t$. This implies that t is the common multiple of a and b. Consider the other common multiple c such that $a|c$and $b|c$.

According to the second condition, we have $t|c$.This implies that $t\le c$. Therefore, t is the least common multiple of a and b.

Hence, it is proved that t is the least common multiple of a and b if and only if t satisfies the following conditions:

1. $a|t$and $b|t$.
2. If $a|c$and $b|c$, then$t|c$.