Class 9 Maths Ganita Manjari Chapter 11 MCQ – The World of Algorithms

An algorithm gives clear instructions in an order that helps us solve a problem. For example, column addition follows steps involving digits and carries. We can execute an algorithm ourselves; a computer is not necessary.

A conditional step is performed only when a stated condition is satisfied. Here, we add j only when dividing n by j leaves remainder 0. If j does not divide n exactly, the list remains unchanged.

The rightmost digit of a whole number is its units digit. Aligning from the right places units below units, tens below tens, and so on. This ensures that we add quantities having the same place value.

Ten units of any place value form one unit of the next higher place value. Therefore, we write 0 in the current column and carry 1 to the left. Carrying is needed when the total is 10 or more.

Each of the two digits can be at most 9, and the incoming carry can be at most 1. Therefore, the largest total is 9 + 9 + 1 = 19. This also explains why the outgoing carry cannot exceed 1.

In the units column, 9 + 1 gives 10. Carrying then makes both the tens and hundreds totals 10. The three written digits are 000. The remaining carry supplies the thousands digit, so the correct answer is 1000.

Writing zeros before a whole number does not change its value. Writing 321 as 00321 makes the place-value columns clear. We can then add every column, including those containing only a digit from 12345, to obtain 12666.

Align the decimal points and write 3.6 as 3.60. Add hundredths first: 5 + 0 = 5. The tenths total is 13, so write 3 and carry 1. Completing the whole-number columns gives 16.35.

Two 3-digit numbers have three input columns, while two 5-digit numbers have five. Therefore, only two additional columns need to be processed. This shows why digit-by-digit addition is more practical than representing large numbers with individual dots and counting them.

A divisor divides the number without leaving a remainder. The factor pairs of 18 are (1, 18), (2, 9) and (3, 6). Collecting both members of each pair and arranging them in increasing order gives [1, 2, 3, 6, 9, 18].

Among the checked numbers 1, 2, 3, 4 and 5, only 1, 2 and 3 divide 18 exactly. Checking 4 and 5 does not change the list. Although 6 is a divisor, it has not yet been checked.

There are 135 integers from 1 through 135, including both endpoints. The algorithm checks each one, whether or not it is a divisor. Therefore, it makes 135 checks, even though the final divisor list contains only eight numbers.

The values checked are 1, 2, 3, 4, 5 and 6 because 6² = 36 and 7² > 36. Each successful check gives the pair j and 36 ÷ j. The repeated member in the pair (6, 6) is listed only once.

The prime numbers that divide 60 are 2, 3 and 5. The number 1 is not prime, and 6 is composite. A list of distinct prime divisors includes 2 only once, even though 60 contains two factors of 2.

A common divisor must appear in both lists. The numbers 1, 2, 3 and 6 satisfy this condition. The numbers 4 and 12 appear only in the first list, while 9 and 18 appear only in the second.

GCD means greatest common divisor. In an increasing list, each later entry is larger than the earlier entries. Therefore, the rightmost entry is the greatest common divisor. For example, the list [1, 2, 3, 6] gives a GCD of 6.

The candidates are checked from largest to smallest. Therefore, the first one that divides both numbers must be their greatest common divisor. Here, that number is 6. Continuing to replace it with smaller common divisors would lose the correct answer.

The list is not in increasing order, so its final entry need not be the greatest. Comparing all its entries gives 6 as the largest. Thus, the GCD is 6, whereas taking the last entry would incorrectly give 3.

A positive divisor of a positive integer cannot be larger than that integer. A common divisor must divide both m and n, so it cannot exceed the smaller number. Checking values above min(m, n) is therefore unnecessary.

Up to 8, the common divisors of 24 and 36 are 1, 2, 3, 4 and 6. The stored value becomes 6 when k = 6. Neither 7 nor 8 divides both numbers, so it remains 6.

Check each entry of List 1 against List 2. The entries 1 and 3 occur in both lists, so they are excluded. The entries 2 and 6 are absent from List 2, so the required result is [2, 6].

The positive multiples of 4 begin 4, 8, 12, 16, while those of 6 begin 6, 12, 18, 24. Their first common entry is 12. Since the lists increase, this is the least positive common multiple, or LCM.

Here, m = 48 and n = 18, so m − n = 30. Substituting into gcd(n, m − n) gives gcd(18, 30). This replacement preserves the common divisors and hence the GCD; the numbers may be reordered in the next step.

The expression 33 mod 7 means the remainder when 33 is divided by 7. Since 33 = 7 × 4 + 5, the remainder is 5. The value 4 is the quotient, not the remainder.

Divide 84 by 30: 84 = 30 × 2 + 24. The algorithm replaces the original pair with the previous divisor and the remainder. Therefore, the next problem is gcd(30, 24), which has the same GCD as the original pair.

First, 825 = 375 × 2 + 75. Next, 375 = 75 × 5 + 0. When the remainder becomes 0, the divisor in that final division is the GCD. Hence, the greatest common divisor is 75.

The stopping rule is gcd(m, 0) = m for positive m. Every positive divisor of 23 also divides 0, and the greatest of these divisors is 23. The algorithm stops here instead of attempting to divide by 0.

Since 99 mod 2 = 1, the first replacement gives gcd(2, 1). Since 2 mod 1 = 0, the second gives gcd(1, 0). Thus, only two remainder replacements are needed, illustrating how division can replace many repeated subtractions.

A prime number has exactly two distinct positive divisors: 1 and itself. Merely having 1 or n in the list does not establish primality because every positive integer has these divisors. The number 1 has only one divisor and is not prime.

To find the GCD, keep only the prime factors shared by both numbers and use the smaller power of each. Both numbers contain 2² and 3², but neither 5 nor 7 is shared. Therefore, the GCD is 2² × 3² = 36.