If a number C leaves a remainder of 4 when di… | GMAT Sprint

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PS · Problem Solving

605-655

Divisibility/Multiples/Factors

If a number C leaves a remainder of 4 when divided by 7, and another number D leaves a remainder of 10 when divided by 15, what is one of the remainders when $C^2 + D^2$ is divided by 15?

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Official Explanation

Official

We are given that

C \equiv 4 \pmod{7}

and

D \equiv 10 \pmod{15}

.

We wish to find the remainder when

C^2 + D^2

is divided by 15.

Step 1. Evaluate

D^2 \pmod{15}

Since

D \equiv 10 \pmod{15}

, we have

D^2 \equiv 10^2 \equiv 100 \pmod{15}.

Express 100 in terms of 15:

100 = 15 \times 6 + 10 \quad \Longrightarrow \quad 100 \equiv 10 \pmod{15}.

Thus,

D^2 \equiv 10 \pmod{15}.

Step 2. Evaluate

C^2 \pmod{15}

We know that

C \equiv 4 \pmod{7}

. This condition does not uniquely determine

C \pmod{15}

, so we find the possible values for

C \pmod{15}

that satisfy

C \equiv 4 \pmod{7}.

Within the range

0 \le r \le 14

, the residues that satisfy

r \equiv 4 \pmod{7}

are:…

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Comments

PaoloFeb 8, 2025

Great question!

SujoyOct 1, 2024

good question

PrasidhamJan 22, 2025

Thank you very much!

What this question tests

What it tests

Your understanding of divisibility rules, least common multiples, and greatest common factors.

Common trap

Forgetting that 1 is not prime and that every integer divides 0.

If a number C leaves a remainder of 4 when divided by 7, and another number D leaves a remainder of 10 when divided by 15, what is one of the remainders when C2+D2C^2 + D^2C2+D2 is divided by 15?