College

Which of the following will leave a remainder of 59?

A. [tex]$1,936 \div 94$[/tex]

B. [tex]$1,937 \div 94$[/tex]

C. [tex]$1,938 \div 94$[/tex]

D. [tex]$1,939 \div 94$[/tex]

Answer :

To solve the problem, we need to determine which division leaves a remainder of [tex]$59$[/tex]. We are given four divisions:

[tex]$$1936 \div 94,\quad 1937 \div 94,\quad 1938 \div 94,\quad \text{and} \quad 1939 \div 94.$$[/tex]

The remainder when a number [tex]$N$[/tex] is divided by [tex]$94$[/tex] is given by

[tex]$$N = 94 \times q + r,$$[/tex]

where [tex]$q$[/tex] is the quotient (an integer) and [tex]$r$[/tex] is the remainder (with [tex]$0 \le r < 94$[/tex]).

Let's go step by step.

1. For [tex]$1936 \div 94$[/tex], let the remainder be [tex]$r_1$[/tex]. When we perform the division, we find:
[tex]$$r_1 = 56.$$[/tex]
(This means [tex]$1936 = 94 \times q_1 + 56$[/tex] for some integer [tex]$q_1$[/tex].)

2. For [tex]$1937 \div 94$[/tex], let the remainder be [tex]$r_2$[/tex]. When we perform the division, we obtain:
[tex]$$r_2 = 57.$$[/tex]
(So [tex]$1937 = 94 \times q_2 + 57$[/tex] for some integer [tex]$q_2$[/tex].)

3. For [tex]$1938 \div 94$[/tex], let the remainder be [tex]$r_3$[/tex]. When we perform the division, we find:
[tex]$$r_3 = 58.$$[/tex]
(Thus [tex]$1938 = 94 \times q_3 + 58$[/tex] for some integer [tex]$q_3$[/tex].)

4. For [tex]$1939 \div 94$[/tex], let the remainder be [tex]$r_4$[/tex]. When we perform the division, we have:
[tex]$$r_4 = 59.$$[/tex]
(So [tex]$1939 = 94 \times q_4 + 59$[/tex] for some integer [tex]$q_4$[/tex].)

Since we are looking for the division that yields a remainder of [tex]$59$[/tex], we see that:

[tex]$$r_4 = 59.$$[/tex]

Thus, the division that leaves a remainder of [tex]$59$[/tex] is

[tex]$$1939 \div 94.$$[/tex]

Therefore, the answer is [tex]$\boxed{1939}$[/tex].