What is the largest two-digit positive integer n such that when n is divided by 4, the rem… — GRE General Test

What is the largest two-digit positive integer $n$ such that when $n$ is divided by $4$ , the remainder is $3$ , and when $n$ is divided by $5$ , the remainder is $2$ ?

Answer: 87

Answer

The largest two-digit positive integer satisfying both remainder conditions is 87.

Any integer satisfying both remainder requirements must be of the form

n = 20m + 7

for an integer

m

. Testing values for

m

shows that

m = 4

yields

n = 87

, which is the largest two-digit integer fitting this rule.

Step-by-Step Solution

Set up modular arithmetic equations for the given remainder conditions.

n \equiv 3 \pmod 4

and

n \equiv 2 \pmod 5

Translate the verbal description of remainders into mathematical congruence relations.

Combine the congruence relations to find the general form of

n

n = 20m + 7

for non-negative integers

m

The least common multiple of

4

and

5

20

, meaning solutions repeat every

20

units.

Find the maximum integer

m

that produces a two-digit integer.

For

m = 4

n = 87

. For

m = 5

n = 107

Two-digit integers are strictly less than

100

Key Concept

Simultaneous Remainders and Divisibility Cycles

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic