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Chapter 1 of 12 • Page 1 of 248🔒 Protected PDF • Watermarked
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Which of the following numbers is a prime number?
119
143
167
187
167
Use unit‑digit and digital‑sum rules to eliminate 2,5,3 quickly, then check primes up to the integer part of the square root.
Numbers to test: 119, 143, 167, 187
n
Use unit‑digit and digital‑sum rules to eliminate 2,5,3 quickly, then check primes up to the integer part of the square root.
Stopping the divisibility check before reaching the integer square‑root limit, which can miss a factor.
Apply basic divisibility rules
Eliminate even numbers and those ending in 5; use the sum of digits to rule out multiples of 3. Thus 119, 143, 167, 187 are all odd and not multiples of 5 or 3.
nmod2=0,nmod5=0,digit_sum(n)mod3=0
Find the upper bound for testing
Compute the integer part of the square root of each candidate. For 167, ⌊167⌋=12, so we only need to test primes ≤12.
⌊167⌋=12
Test divisibility by primes ≤12
Check 7, 11 for each number. 119 ÷ 7 = 17, 143 ÷ 11 = 13, 187 ÷ 11 = 17, while 167 is not divisible by 7 or 11, confirming it has no factors up to 12.
167mod7=0,167mod11=0
Determine the prime
Since 167 has no divisor ≤12, it is prime; the others are composite.
167 is prime
C is correct because 167 has no divisors other than 1 and itself, while the other options are composite.
Understanding the n bound is also useful in simplifying algebraic factorisation and in solving Diophantine equations in number‑theory problems.