Let a be a number written (in base 10) as a_0 ·10^0 +a_1 ·10^1 +a_2 ·10^2 +···+a

Question

Let a be a number written (in base 10) as

a_0 ·10^0 +a_1 ·10^1 +a_2 ·10^2 +···+a_n ·10^n where 0 a_i < 10.

(i) Prove that 2 divides a if and only if 2 divides a_0.

(ii) Prove that 4 divides a if and only if 4 divides a_0 +2a_1.

(iii) Prove that 8 divides a if and only if 8 divides a_0 +2a_1 +4a_2.

(iv) Prove that 5 divides a if and only if 5 divides a_0.

(v) Prove that 9 divides a if and only if 9 divides the sum a_0 +a_1 +···+a_n of its digits.

(vi) Prove that 3 divides a if and only if 3 divides the sum of its digits.

(vii) Prove that 11 divides a if and only if 11 divides a_0 a_1 +a_2 ···.

Explanation / Answer

(According to Chegg policy, only four subquestions will be answered. Please post the remaining in another question)

a = a0 ·100 + a1 ·101 +a2 ·102 +···+an ·10n

(i) a = a0 ·100 + a1 ·101 +a2 ·102 +···+an ·10n

=> a = a0 ·100 + 10 (a1 +a2 ·10 +···+an ·10n-1)

Since 2 | 10, 2 | 10 (a1 +a2 ·10 +···+an ·10n-1)

Let 10 (a1 +a2 ·10 +···+an ·10n-1) = 2k

=> a = a0 + 2k

If 2 | a0, let a0 = 2r => a = 2r + 2k = 2 (r + k) => 2 | a

If 2 | a, let a = 2s => 2s = a0 + 2k => a0 = 2s - 2k = 2 (s - k) => 2 | a0

(ii) a = a0 ·100 + a1 ·101 +a2 ·102 +···+an ·10n

=> a = a0 ·100 + a1 ·101 + 102 (a2 +a3 ·10 +···+an ·10n-2)

Since 4 | 102, 4 | 102 (a2 +a3 ·10 +···+an ·10n-2)

Let 102 (a2 +a3 ·10 +···+an ·10n-2) = 4j

=> a = a0 ·100 + a1 ·101 + 4j = a0 + 2a1 + 8a1 + 4j = a0 + 2a1 + 4(2a1 + j) = a0 + 2a1 + 4k, where k = 2a1 + j

If 4 | a0 + 2a1, let a0 + 2a1 = 4r => a = 4r + 4k = 4 (r + k) => 4 | a

If 4 | a, let a = 4s => 4s = a0 + 2a1 + 4k => a0 + 2a1 = 4s - 4k = 4 (s - k) => 4 | a0 + 2a1

(iii) a = a0 ·100 + a1 ·101 + a2 ·102 +···+an ·10n

=> a = a0 ·100 + a1 ·101 + a2 ·102 + 103 (a3 + a4 ·10 +···+an ·10n-3)

Since 8 | 103, 8 | 103 (a3 + a4 ·10 +···+an ·10n-3)

Let 103 (a3 + a4 ·10 +···+an ·10n-3) = 8j

=> a = a0 ·100 + a1 ·101 + a2 ·102 + 8j = a0 ·100 + 2a1 + 8a1+ 4a2 + 96a2 + 8j = a0 ·100 + 2a1 + 4a2 + 8(a1 + 12a2 + j) = a0 ·100 + 2a1+ 4a2 + 8k, where k = 12a2 + j

If 8 | a0 ·100 + 2a1+ 4a2, let a0 ·100 + 2a1+ 4a2 = 8r => a = 8r + 8k = 8 (r + k) => 8 | a

If 8 | a, let a = 8s => 8s = a0 ·100 + 2a1+ 4a2 + 8k => a0 ·100 + 2a1+ 4a2 = 8s - 8k = 8 (s - k) => 8 | a0 ·100 + 2a1+ 4a2.

(iv) a = a0 ·100 + a1 ·101 +a2 ·102 +···+an ·10n

=> a = a0 ·100 + 10 (a1 +a2 ·10 +···+an ·10n-1)

Since 5 | 10, 5 | 10 (a1 +a2 ·10 +···+an ·10n-1)

Let 10 (a1 +a2 ·10 +···+an ·10n-1) = 5k

=> a = a0 + 5k

If 5 | a0, let a0 = 5r => a = 5r + 5k = 5 (r + k) => 5 | a

If 5 | a, let a = 5s => 5s = a0 + 5k => a0 = 5s - 5k = 5 (s - k) => 5 | a0

Related Questions

Navigate

• Browse (All)
• Browse L
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.