Remainder Theorem Level - 1
Q1–10 of 50Find the remainder
\( \dfrac{1789\times1790\times1791}{1788} \)
Step 1: Find individual remainders modulo $1788$:
$1789 \equiv +1 \pmod{1788}$
$1790 \equiv +2 \pmod{1788}$
$1791 \equiv +3 \pmod{1788}$
Step 2: Multiply remainders: $1 \times 2 \times 3 = 6$.
Correct Option: (d) 6
Find the remainder :
\( \dfrac{3^{100}}{2} \)
Step 1: $3 \equiv 1 \pmod 2$.
Step 2: $3^{100} \equiv 1^{100} = 1 \pmod 2$.
Answer: 1
Find the remainder :
\( \dfrac{3^{100}+5}{2} \)
Step 1: From Q2, $3^{100} \equiv 1 \pmod 2$.
Step 2: $5 \equiv 1 \pmod 2$.
Step 3: Remainder $= 1 + 1 = 2 \equiv 0 \pmod 2$.
Find the remainder :
\( \dfrac{18^{75}+9}{2} \)
Step 1: $18 \equiv 0 \pmod 2 \implies 18^{75} \equiv 0 \pmod 2$.
Step 2: $9 \equiv 1 \pmod 2$.
Step 3: Remainder $= 0 + 1 = 1$.
Answer
Direction for this questions: If answer is 0 type it as zero, if answer is 1 type it as one and so on.
Find the remainder :
\( \dfrac{5^4-1}{4} \)
Step 1: $5 \equiv 1 \pmod 4 \implies 5^4 \equiv 1^4 = 1 \pmod 4$.
Step 2: Remainder $= 1 - 1 = 0$.
Answer: 0
Find the remainder :
\( \dfrac{97^{89}+87}{96} \)
Step 1: $97 \equiv +1 \pmod{96} \implies 97^{89} \equiv 1^{89} = 1 \pmod{96}$.
Step 2: $87 \equiv 87 \pmod{96}$.
Step 3: Total remainder $= 1 + 87 = 88$.
Find the remainder :
\( \dfrac{1789\times1790}{1791} \)
Step 1: Express terms using negative remainders:
$1789 \equiv -2 \pmod{1791}$
$1790 \equiv -1 \pmod{1791}$
Step 2: Multiply: $(-2) \times (-1) = 2$.
Find the remainder :
\( \dfrac{1764\times1765\times1766\times1767}{1768} \)
Step 1: Use negative remainders modulo $1768$:
$1764 \equiv -4$, $1765 \equiv -3$, $1766 \equiv -2$, $1767 \equiv -1$.
Step 2: Multiply: $(-4) \times (-3) \times (-2) \times (-1) = 24$.
Answer: 24
Find the remainder :
\( \dfrac{1764\times1765\times1766}{1768} \)
Step 1: Negative remainders modulo $1768$:
$1764 \equiv -4$, $1765 \equiv -3$, $1766 \equiv -2$.
Step 2: Multiply: $(-4) \times (-3) \times (-2) = -24$.
Step 3: Convert to positive remainder: $1768 - 24 = 1744$.
Correct Option: (b) 1744
Find the remainder :
\( \dfrac{2^{101}-1}{3} \)
Step 1: $2 \equiv -1 \pmod 3$.
Step 2: $2^{101} \equiv (-1)^{101} = -1 \pmod 3$.
Step 3: $(-1 - 1) = -2 \equiv 1 \pmod 3$.
Answer: 1