What is the least positive integer that satisfies the following conditions?

a) When divided by 2, the remainder is 1.

b) When divided by 3, the remainder is 2.

c) When divided by 4, the remainder is 3.

d) When divided by 5, the remainder is 4.
Let the smallest common solution be $a$. The given system of congruences is \begin{align*}
a\equiv 1\equiv -1\pmod 2,\\
a\equiv 2\equiv -1\pmod 3,\\
a\equiv 3\equiv -1\pmod 4,\\
a\equiv 4\equiv -1\pmod 5.
\end{align*} Note that if $a\equiv-1\pmod 4$, then $a\equiv-1\pmod 2$ as well,  so we need only consider the final three congruences. Since $\gcd(3,4)=\gcd(4,5)=\gcd(3,5)=1$, we have $$a\equiv -1\pmod{3\cdot 4\cdot 5},$$ that is, $a\equiv 59\pmod{60}$.

So $a$ has a lower bound of $59$, but $59$ also happens to satisfy all of the original congruences. Thus, $a=\boxed{59}$.
59
Level 3
Number Theory
Problem: What is the least positive integer that satisfies the following conditions?

a) When divided by 2, the remainder is 1.

b) When divided by 3, the remainder is 2.

c) When divided by 4, the remainder is 3.

d) When divided by 5, the remainder is 4.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the least positive integer that meets the specified conditions, we can express each condition mathematically. 

The conditions can be rewritten as a system of congruences:
1. x ≡ 1 (mod 2)
2. x ≡ 2 (mod 3)
3. x ≡ 3 (mod 4)
4. x ≡ 4 (mod 5)

Next, we observe that each condition indicates that the integer x is one less than the modulus of each divisor. This suggests a pattern where x can be expressed in terms of the least common multiple (LCM) of the divisors plus a constant.

To solve the system, we can start with the largest modulus, which is 5. We can express x in the form:
x = 5k + 4 for some integer k.

Now, substituting this expression into the third condition:
5k + 4 ≡ 3 (mod 4)
This simplifies to:
k + 0 ≡ 3 (mod 4)
Thus, k ≡ 3 (mod 4), which means:
k = 4m + 3 for some integer m.

Substituting back into the expression for x gives:
x = 5(4m + 3) + 4 = 20m + 15 + 4 = 20m + 19.

Next, we substitute this into the second condition:
20m + 19 ≡ 2 (mod 3)
This simplifies to:
2m + 1 ≡ 2 (mod 3)
Thus, 2m ≡ 1 (mod 3), which leads to:
m ≡ 2 (mod 3), meaning:
m = 3n + 2 for some integer n.

Substituting back into the expression for x gives:
x = 20(3n + 2) + 19 = 60n + 40 + 19 = 60n + 59.

To find the least positive integer, we set n = 0:
x = 59.

Thus, the least positive integer that satisfies all the conditions is 59.<|END_OF_LATENT|> The final answer is $59$.