Number System for the Assam Police Constable Exam
The Number System is the first chapter of Elementary Mathematics in the SLPRB Assam Police Constable (AB & UB) written test, and it quietly supports every chapter that follows. Percentage, ratio, profit and loss, and time and work all begin with a number, and a candidate who is comfortable with place value, divisibility, fractions and roots reads those questions faster and makes fewer slips. In the Assam Police paper the number system is rarely asked as one long sum. Instead, short objective questions appear on divisibility, on the smallest or largest number of a certain type, on the unit digit of a large power, on the number of factors of a number, and on the correct order of a mixed set of values. Each of these takes only a few seconds when the rule is known, which makes this chapter one of the quickest places to collect marks and the safest place to protect the score from negative marking. The approach followed here matches the Master Book: Maths Masti (D. Hurina): keep the number line and the place-value chart in view, and convert every question into a familiar family of numbers before any calculation begins.
Why This Chapter Matters in the Assam Police Paper
The elementary mathematics section rewards speed and accuracy in equal measure, and the number system supplies both. It turns a long-looking question into the application of one fixed rule. To find the smallest four-digit number divisible by nine, a candidate only applies the divisibility rule for nine and adjusts one digit. To find the unit digit of a large power, the candidate cycles the last digits and reads off the answer. To find the number of divisors, the candidate writes the prime factorisation and multiplies one more than each exponent. None of these needs lengthy working, and all of them rest on the ideas collected in this chapter. The same ideas also keep the later chapters clean, because a wrong simplification at the start of a percentage or ratio sum is usually a number-system mistake in disguise.
Terms You Should Know
- Natural numbers: the counting numbers 1, 2, 3 and so on.
- Whole numbers: the natural numbers together with zero.
- Integers: whole numbers and their negatives.
- Rational number: a number that can be written as a fraction of two integers with a non-zero denominator.
- Irrational number: a number that cannot be written as such a fraction, for example the square root of two.
- Prime number: a number greater than one having exactly two factors, one and itself.
- Composite number: a number greater than one having more than two factors.
- Co-prime numbers: two numbers whose highest common factor is one.
Two primes that differ by two, such as eleven and thirteen, are called twin primes. The number one is neither prime nor composite. Two is the only even prime, and this single fact settles several objective questions on its own. Zero is neither positive nor negative, and it is the only number that is neither prime nor composite and also neither positive nor negative.
Families of Numbers
Even numbers are divisible by two, and odd numbers leave a remainder of one when divided by two. A number is a multiple of three when the sum of its digits is a multiple of three. Squares end only in 0, 1, 4, 5, 6 or 9, so a number ending in 2, 3, 7 or 8 can never be a perfect square. Cubes may end in any digit, but a cube ending in 8 must have a root ending in 2, and a cube ending in 7 must have a root ending in 3. Recognising these families lets a candidate reject wrong options without doing the full sum, which is exactly the skill the objective paper rewards.
Place Value, Face Value and Number Formation
The face value of a digit is the digit itself, while its place value is the digit multiplied by the value of its position. In the number 4,07,536 the digit seven has a face value of seven and a place value of seven thousand. The largest number formed from a given set of digits is made by writing them in descending order, and the smallest by writing them in ascending order, remembering that a number cannot begin with zero. These two rules answer a large family of questions on the greatest and smallest number of a given number of digits.
Divisibility Rules That Save Time
A number is divisible by two if its last digit is even, by five if it ends in zero or five, and by ten if it ends in zero. It is divisible by four if the last two digits form a multiple of four, and by eight if the last three digits form a multiple of eight. It is divisible by three if the digit sum is a multiple of three and by nine if the digit sum is a multiple of nine. It is divisible by eleven if the alternating sum of its digits is zero or a multiple of eleven. For six, twelve and fifteen, test the two factors separately, remembering that the factors must be co-prime for this to work, so twelve is tested by three and four, and fifteen by three and five.
Prime Factorisation, HCF and LCM
Every composite number can be broken into a product of primes in only one way. The highest common factor is the product of the common primes taken to their lowest power, and the lowest common multiple is the product of all the primes taken to their highest power. For two numbers, the product of the highest common factor and the lowest common multiple equals the product of the numbers themselves, a relation that lets a candidate find one from the other in a single step. Co-prime numbers have a highest common factor of one, and their lowest common multiple is simply their product.
Fractions, Decimals and Ordering
To compare fractions, cross-multiply, or convert each to a decimal, or give them a common denominator; the largest numerator over the same denominator is the largest fraction. A decimal can always be written as a fraction by counting the number of places after the point and dividing by the matching power of ten. Terminating decimals and recurring decimals both come from fractions, and converting a recurring decimal to a fraction is a standard short question.
Squares, Cubes and Roots
The squares of numbers from one to thirty and the cubes of numbers from one to fifteen should be memorised, because they appear as hidden steps in many sums. The square root of a perfect square can be found by pairing the digits from the right, and the number of digits in the square root is half the number of digits in the square, rounded up. The cube root of a perfect cube is found by splitting the number into groups of three digits from the right and using the fact that each cube ending decides its last digit.
Patterns Asked in Assam Police Papers
The most common patterns are: the unit digit of a product or a power; the smallest or largest number of a given number of digits satisfying a divisibility condition; the number of factors or the sum of factors of a number; the greatest number that divides several given numbers leaving fixed remainders; and the arrangement of fractions or decimals in increasing order. A question on the digit in a particular place of a large product, and a question on the remainder when a number is divided by nine, also appear often. Because each pattern has a fixed route, practising the patterns is more useful than practising random sums.
Shortcut Techniques for the Exam Hall
For the unit digit, take only the last digit of the base to its power and reduce the exponent modulo the cycle length, which is four for most digits. For divisibility by nine or three, add the digits and repeat until one digit remains. For the number of factors, add one to each exponent in the prime factorisation and multiply. For the largest number leaving the same remainder, take the highest common factor of the differences. For ordering a mixed set, convert everything to decimals of the same length. These shortcuts keep the working to one or two lines and reduce careless errors.
Common Traps and Errors
- Treating one as a prime number; it is neither prime nor composite.
- Forgetting that the smallest number cannot begin with a zero digit.
- Testing divisibility by twelve with four and six, which are not co-prime, instead of three and four.
- Confusing the face value of a digit with its place value.
- Assuming a number ending in two, three, seven or eight is a perfect square.
- Using the product-of-numbers rule between the highest common factor and the lowest common multiple for more than two numbers.
A Study Plan for This Chapter
- Revise the classes of numbers and the meaning of place and face value.
- Memorise the divisibility rules from two to eleven and practise them on random numbers.
- Learn prime factorisation and the highest common factor and lowest common multiple together.
- Drill the unit-digit cycle and the squares and cubes up to the recommended limits.
- Finish with a mixed set of objective questions and revise the traps each week.
Frequently Asked Questions
Is one a prime number?
No. A prime number must have exactly two distinct factors, and the number one has only one factor, so it is neither prime nor composite.
How do I find the unit digit of a large power quickly?
Take only the last digit of the base, divide the exponent by four, and use the remainder, because the last digits of most bases repeat in a cycle of four. A remainder of one gives the base digit itself, two gives its square, three gives its cube, and a remainder of zero gives the fourth power.
What is the fastest way to count the factors of a number?
Write the prime factorisation, add one to each exponent, and multiply these numbers. For example, 72 is two cubed times three squared, and the number of factors is four times three, which is twelve.
Why is the product of the HCF and LCM equal to the product of the numbers?
Because the highest common factor takes the common primes at their lowest power and the lowest common multiple takes all primes at their highest power, so together they count each prime exactly as often as the two numbers do. This rule holds only for two numbers.
How do I find the greatest number that divides several numbers leaving the same remainder?
Subtract the remainder from each number and take the highest common factor of the differences. The result is the greatest number that leaves that common remainder.