Divisibility and Multiples | FIO

Question 15

What will be the digital root of the number $9a + 36b + 13$ ?

Solution

The digital root of a number is its remainder when divided by 9 (if the remainder is not 0), or 9 (if the remainder is 0).

Step 1 — Understand Digital Root

The digital root of a number is the single digit you get by repeatedly adding its digits. For example, the digital root of $25$ is $2+5=7$ . The digital root of $48$ is $4+8=12$ , then $1+2=3$ . A key property is that the digital root of a number is the same as its remainder when divided by 9. We use the notation "modulo 9" or "mod 9" for this remainder. For example, $25 \div 9$ gives a remainder of $7$ . So, we write $25 \equiv 7 \pmod 9$ . If a number is a multiple of 9, like 18, its remainder when divided by 9 is 0. However, its digital root is $1+8=9$ . So, if the remainder is 0, the digital root is 9. We will find the digital root of the expression by finding its value modulo 9.

Step 2 — Simplify each term modulo 9

We need to find the digital root of the expression $9a + 36b + 13$ . This means we need to find the value of $(9a + 36b + 13) \pmod 9$ . Let us look at each term in the expression separately.

First, consider the term $9a$ . Any number that is a multiple of 9 will have a remainder of 0 when divided by 9. $9a \equiv 0 \pmod 9$

Next, consider the term $36b$ . The number $36$ is a multiple of $9$ , because $36 = 4 \times 9$ . So, $36b$ will also be a multiple of $9$ . $36b \equiv 0 \pmod 9$

Finally, consider the constant term $13$ . We divide $13$ by $9$ to find its remainder. $13 = 1 \times 9 + 4$ So, the remainder is $4$ . $13 \equiv 4 \pmod 9$

Step 3 — Combine the remainders

Now we add the remainders we found for each term. The digital root of the entire expression will be the digital root of the sum of these remainders. $(9a + 36b + 13) \pmod 9 \equiv (0 + 0 + 4) \pmod 9$ $\equiv 4 \pmod 9$ The remainder when $9a + 36b + 13$ is divided by $9$ is $4$ . Since the remainder is not $0$ , the digital root is this remainder.

$\boxed{4}$

Answer

The digital root of the number $9a + 36b + 13$ is 4.

More questions in FIO

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For each statement below, determine whether it is always true, sometimes true, or never true. Explain your answer. Mention examples and non-examples as appropriate. Justify your claim using algebra.

(i) The sum of two even numbers is a multiple of 3.

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(iii) If two numbers are not divisible by 6, then their sum is not divisible by 6.

(iv) The sum of a multiple of 6 and a multiple of 9 is a multiple of 3.

(v) The sum of a multiple of 6 and a multiple of 3 is a multiple of 9.

Find a few numbers that leave a remainder of 2 when divided by 3 and a remainder of 2 when divided by 4. Write an algebraic expression to describe all such numbers.

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Find, without dividing, whether the following numbers are divisible by 9.

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Q10

Find the smallest multiple of 9 with no odd digits.

Q11

Find the multiple of 9 that is closest to the number 6000.

Q12

How many multiples of 9 are there between the numbers 4300 and 4400?

Q13

The digital root of an 8-digit number is 5. What will be the digital root of 10 more than that number?

Q14

Write any number. Generate a sequence of numbers by repeatedly adding 11. What would be the digital roots of this sequence of numbers? Share your observations.

Q15

What will be the digital root of the number $9a + 36b + 13$ ?

Q16

Make conjectures by examining if there are any patterns or relations between (i) the parity of a number and its digital root. (ii) the digital root of a number and the remainder obtained when the number is divided by 3 or 9.

Q17

If 31z5 is a multiple of 9, where z is a digit, what is the value of z? Explain why there are two answers to this problem.

Q18

"I take a number that leaves a remainder of 8 when divided by 12. I take another number which is 4 short of a multiple of 12. Their sum will always be a multiple of 8", claims Snehal. Examine his claim and justify your conclusion.

Q19

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Q20

Sreelatha says, "I have a number that is divisible by 9. If I reverse its digits, it will still be divisible by 9".

(i) Examine if her conjecture is true for any multiple of 9.

(ii) Are any other digit shuffles possible such that the number formed is still a multiple of 9?

Q21

If 48a23b is a multiple of 18, list all possible pairs of values for a and b.

Q22

If $3p7q8$ is divisible by 44, list all possible pairs of values for $p$ and $q$ .

Q23

Find three consecutive numbers such that the first number is a multiple of 2, the second number is a multiple of 3, and the third number is a multiple of 4.

Are there more such numbers? How often do they occur?

Q24

Write five multiples of 36 between 45,000 and 47,000. Share your approach with the class.

Q25

The middle number in the sequence of 5 consecutive even numbers is $5p$ . Express the other four numbers in sequence in terms of $p$ .

Q26

Write a 6-digit number that it is divisible by 15, such that when the digits are reversed, it is divisible by 6.

Q27

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Q28

Determine whether the statements below are 'Always True', 'Sometimes True', or 'Never True'. Explain your reasoning.

(i) The product of a multiple of 6 and a multiple of 3 is a multiple of 9. (ii) The sum of three consecutive even numbers will be divisible by 6. (iii) If $abcdef$ is a multiple of 6, then $badcef$ will be a multiple of 6. (iv) $8 (7b - 3) - 4 (11b + 1)$ is a multiple of 12.

Q29

Choose any 3 numbers. When is their sum divisible by 3? Explore all possible cases and generalise.

Q30

Is the product of two consecutive integers always multiple of 2? Why? What about the product of these consecutive integers? Is it always a multiple of 6? Why or why not? What can you say about the product of 4 consecutive integers? What about the product of five consecutive integers?

Q31

Solve the cryptarithms —

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Q32

Which of the following Venn diagrams captures the relationship between the multiples of 4, 8, and 32?