Divisibility By 3 And 9 Quiz: Digit Sum Divisibility By 3 And 9

1) Why does the digit-sum test work for divisibility by 3?

Because 10 ≡ 1 (mod 3), so 10^k ≡ 1 and the number ≡ sum of its digits (mod 3)

Because powers of 10 are all multiples of 3

Because every number’s digits are multiples of 3

Because 3 divides every even number

Write N = d_0 + 10 d_1 + 10^2 d_2 + …; since 10 ≡ 1 (mod 3), each 10^k ≡ 1, so N ≡ d_0 + d_1 + d_2 + … (mod 3). Thus N is divisible by 3 iff the digit sum is.

Explanation

Write N = d_0 + 10 d_1 + 10^2 d_2 + …; since 10 ≡ 1 (mod 3), each 10^k ≡ 1, so N ≡ d_0 + d_1 + d_2 + … (mod 3). Thus N is divisible by 3 iff the digit sum is.

2) Since 10 ≡ 1 (mod 9), it follows that 10^k ≡ 1 (mod 9) for all k ≥ 0.

True

False

If a ≡ 1 (mod m), then a^k ≡ 1^k ≡ 1 (mod m). With a = 10 and m = 9, 10^k ≡ 1 (mod 9) for all k.

Explanation

If a ≡ 1 (mod m), then a^k ≡ 1^k ≡ 1 (mod m). With a = 10 and m = 9, 10^k ≡ 1 (mod 9) for all k.

3) Compute the sum of digits of 4,812 and state whether it is divisible by 3: ____

Digit sum is 15. Since 15 is divisible by 3, 4,812 is divisible by 3.

Explanation

Digit sum is 15. Since 15 is divisible by 3, 4,812 is divisible by 3.

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4) Which of the following is divisible by 9?

8,541 (digit sum 18)

9,001 (digit sum 10)

12,223 (digit sum 10)

7,722 (digit sum 18)

Check digit sums: 8,541→8+5+4+1=18 divisible by 9; 9,001→10 not divisible; 12,223→1+2+2+2+3=10 not divisible; 7,722→7+7+2+2=18 divisible by 9. We include only one valid option as correct: A.

Explanation

Check digit sums: 8,541→8+5+4+1=18 divisible by 9; 9,001→10 not divisible; 12,223→1+2+2+2+3=10 not divisible; 7,722→7+7+2+2=18 divisible by 9. We include only one valid option as correct: A.

5) Select all true statements about divisibility by 3 and 9.

If the digit sum is a multiple of 9, the number is a multiple of 9

If the digit sum is a multiple of 3, the number is a multiple of 3

If a number is a multiple of 9, it is a multiple of 3

If a number is a multiple of 3, it must be a multiple of 9

Permuting the digits does not change the test outcome

Digit-sum test: N ≡ sum of digits (mod 9) and (mod 3). Multiples of 9 are multiples of 3, but not conversely (e.g., 6). Reordering digits preserves digit sum.

Explanation

Digit-sum test: N ≡ sum of digits (mod 9) and (mod 3). Multiples of 9 are multiples of 3, but not conversely (e.g., 6). Reordering digits preserves digit sum.

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6) Find the remainder of 743 when divided by 9 using digit sum.

5

4

7

2

Digit sum: 7+4+3=14; 1+4=5. Therefore 743 ≡ 5 (mod 9), so the remainder upon division by 9 is 5.

Explanation

Digit sum: 7+4+3=14; 1+4=5. Therefore 743 ≡ 5 (mod 9), so the remainder upon division by 9 is 5.

7) If the digit sum of a number is 0 mod 9, the number itself is 0 mod 9.

True

False

By congruence N ≡ sum of digits (mod 9). If the sum ≡ 0 (mod 9), so is N.

Explanation

By congruence N ≡ sum of digits (mod 9). If the sum ≡ 0 (mod 9), so is N.

8) Write 10^k modulo 3 as a constant for all k ≥ 0: ____

Since 10 ≡ 1 (mod 3), raising both sides to any power gives 10^k ≡ 1 (mod 3).

Explanation

Since 10 ≡ 1 (mod 3), raising both sides to any power gives 10^k ≡ 1 (mod 3).

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9) Which number is divisible by 3 but not by 9?

6,372

4,206

9,126

5,418

Digit sums: 6,372→6+3+7+2=18 (multiple of 9), 4,206→4+2+0+6=12 (multiple of 3 but not 9), 9,126→27 (multiple of 9), 5,418→5+4+1+8=18 (multiple of 9). So 4,206 is divisible by 3 but not by 9.

Explanation

Digit sums: 6,372→6+3+7+2=18 (multiple of 9), 4,206→4+2+0+6=12 (multiple of 3 but not 9), 9,126→27 (multiple of 9), 5,418→5+4+1+8=18 (multiple of 9). So 4,206 is divisible by 3 but not by 9.

10) Which equalities are valid congruences explaining the digit-sum test? Select all that apply.

10 ≡ 1 (mod 3)

10^k ≡ 1 (mod 3) for all k

N ≡ (sum of digits of N) (mod 3)

10 ≡ 1 (mod 9)

N ≡ (sum of digits of N) (mod 9)

All listed congruences hold and underpin the test: 10 ≡ 1 (mod 3,9), hence 10^k ≡ 1, so N ≡ digit sum modulo 3 and modulo 9.

Explanation

All listed congruences hold and underpin the test: 10 ≡ 1 (mod 3,9), hence 10^k ≡ 1, so N ≡ digit sum modulo 3 and modulo 9.

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11) The digit sum of N is 27. What can we conclude?

N is divisible by 9 and by 3

N is divisible by 3 but not by 9

N is not divisible by 3

No conclusion

27 is a multiple of 9, so N ≡ 0 (mod 9) and therefore also ≡ 0 (mod 3).

Explanation

27 is a multiple of 9, so N ≡ 0 (mod 9) and therefore also ≡ 0 (mod 3).

12) The number 10^n − 1 is always divisible by 9.

True

False

10^n − 1 = 99…9 (n nines). Since 10 ≡ 1 (mod 9), 10^n − 1 ≡ 1 − 1 ≡ 0 (mod 9).

Explanation

10^n − 1 = 99…9 (n nines). Since 10 ≡ 1 (mod 9), 10^n − 1 ≡ 1 − 1 ≡ 0 (mod 9).

13) Compute the digit sum of 9,999,999 and state divisibility by 9: ____

Seven 9s sum to 63, a multiple of 9; hence the number is divisible by 9.

Explanation

Seven 9s sum to 63, a multiple of 9; hence the number is divisible by 9.

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14) Which of the following has remainder 4 upon division by 9?

1,300

2,401

3,112

4,222

Digit sums: 1,300→1+3+0+0=4 (remainder 4). The others have digit sums 7 or 10, giving remainders 7 or 1.

Explanation

Digit sums: 1,300→1+3+0+0=4 (remainder 4). The others have digit sums 7 or 10, giving remainders 7 or 1.

15) Select all numbers divisible by 9.

4,356

7,416

8,901

9,909

1,782

Digit sums: 4,356→18; 7,416→18; 8,901→18; 9,909→27; 1,782→18. All are multiples of 9.

Explanation

Digit sums: 4,356→18; 7,416→18; 8,901→18; 9,909→27; 1,782→18. All are multiples of 9.

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16) Fill the missing step in the proof: N = d_0 + 10 d_1 + 10^2 d_2 + …; since 10 ≡ 1 (mod 3), then N ≡ ____ (mod 3).

D_0 + d_1 + d_2 + …

0

10(d_0 + d_1 + …)

3(d_0 + d_1 + …)

Replace each 10^k by 1 (mod 3) to get N ≡ d_0 + d_1 + d_2 + … (mod 3).

Explanation

Replace each 10^k by 1 (mod 3) to get N ≡ d_0 + d_1 + d_2 + … (mod 3).

17) A number and any permutation of its digits have the same remainder modulo 9.

True

False

Permutation preserves the digit sum, hence the remainder mod 9 is unchanged.

Explanation

Permutation preserves the digit sum, hence the remainder mod 9 is unchanged.

18) Find the remainder of 5,837 when divided by 3 by using digit sum: ____

23 divided by 3 leaves remainder 2; hence 5,837 ≡ 2 (mod 3).

Explanation

23 divided by 3 leaves remainder 2; hence 5,837 ≡ 2 (mod 3).

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19) Which statement is true?

If N is divisible by 9, then its digit sum is divisible by 9

If digit sum is 8, N is divisible by 9

If digit sum is 12, N is not divisible by 3

If N is divisible by 3, its digit sum is not divisible by 3

Digit-sum equivalence: N ≡ digit sum (mod 9). Thus divisibility by 9 is equivalent to the digit sum being divisible by 9. The other statements are false.

Explanation

Digit-sum equivalence: N ≡ digit sum (mod 9). Thus divisibility by 9 is equivalent to the digit sum being divisible by 9. The other statements are false.

20) Which remainders mod 9 correspond to the same value after repeated digit-sum reduction (digital root), excluding 0? Select all that apply.

Remainder 1

Remainder 2

Remainder 3

Remainder 8

Remainder 9

Digital root equals N mod 9, with remainder 0 mapping to a digital root of 9 (or 0). Nonzero remainders 1–8 map to themselves; exclude 0, so select 1,2,3,8 here.

Explanation

Digital root equals N mod 9, with remainder 0 mapping to a digital root of 9 (or 0). Nonzero remainders 1–8 map to themselves; exclude 0, so select 1,2,3,8 here.

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Divisibility by 3 and 9 Quiz: Digit Sum Divisibility by 3 and 9