What is 10^{40}\mod 210 ?

The solution to 10^{40}\mod 210 is 130

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Popular >

10^{40}\mod 210

Solution

1 0^{40} mod 210

Solution

130

Solution steps

1 0^{40} mod 210

1 0^{40} = (1 0^{3})^{13} \cdot 10

1 0^{40}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 1 0^{39} \cdot 10

Apply exponent rule:

a^{b c} = (a^{b})^{c}

= (1 0^{3})^{13} \cdot 10

= (1 0^{3})^{13} \cdot 10 mod 210

Apply modulo rule:

(a \cdot b) mod c = ((a mod c) \cdot (b mod c)) mod c

= (((1 0^{3})^{13} mod 210) \cdot (10 mod 210)) mod 210

Apply modulo rule:

(a^{b}) mod c = ((a mod c)^{b}) mod c

= ((((1 0^{3} mod 210)^{13}) mod 210) \cdot (10 mod 210)) mod 210

10 mod 210 : 10

b > a > 0

then

a mod b = a

= 10

= ((((1 0^{3} mod 210)^{13}) mod 210) \cdot 10) mod 210

1 0^{3} = 1000

= ((((1000 mod 210)^{13}) mod 210) \cdot 10) mod 210

1000 mod 210 : 160

Find Remainder:

Long Division

\frac{1000}{210} : 4

Remainder

160

\frac{1000}{210}

Write the problem in long division format

210 \overline{∣ 1000}

Divide

1000

210

to get

4

Divide

1000

210

to get

4

4 210 \overline{∣ 1000}

Multiply the quotient digit

(4)

by the divisor

210

4 210 \overline{∣ 1000} \underline{840}

Subtract

840

from

1000

4 210 \overline{∣ 1000} \underline{840} 160

4 210 \overline{∣ 1000} \underline{840} 160

The solution for Long Division of

\frac{1000}{210}

4

with remainder of

160

4 Remainder 160

= 160

= ((16 0^{13} mod 210) \cdot 10) mod 210

16 0^{13} = (16 0^{2})^{6} \cdot 160

16 0^{13}

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= 16 0^{12} \cdot 160

Apply exponent rule:

a^{b c} = (a^{b})^{c}

= (16 0^{2})^{6} \cdot 160

= (((16 0^{2})^{6} \cdot 160 mod 210) \cdot 10) mod 210

Apply modulo rule:

(a \cdot b) mod c = ((a mod c) \cdot (b mod c)) mod c

= (((((16 0^{2})^{6} mod 210) \cdot (160 mod 210)) mod 210) \cdot 10) mod 210

Apply modulo rule:

(a^{b}) mod c = ((a mod c)^{b}) mod c

= ((((((16 0^{2} mod 210)^{6}) mod 210) \cdot (160 mod 210)) mod 210) \cdot 10) mod 210

160 mod 210 : 160

b > a > 0

then

a mod b = a

= 160

= ((((((16 0^{2} mod 210)^{6}) mod 210) \cdot 160) mod 210) \cdot 10) mod 210

16 0^{2} = 25600

= ((((((25600 mod 210)^{6}) mod 210) \cdot 160) mod 210) \cdot 10) mod 210

25600 mod 210 : 190

Find Remainder:

Long Division

\frac{25600}{210} : 121

Remainder

190

\frac{25600}{210}

Write the problem in long division format

210 \overline{∣ 25600}

Divide

256

210

to get

1

Divide

256

210

to get

1

1 210 \overline{∣ 25600}

Multiply the quotient digit

(1)

by the divisor

210

1 210 \overline{∣ 25600} \underline{210}

Subtract

210

from

256

1 210 \overline{∣ 25600} \underline{210} 46

Bring down the next number of the dividend

1 210 \overline{∣ 25600} \underline{210} 460

1 210 \overline{∣ 25600} \underline{210} 460

Divide

460

210

to get

2

Divide

460

210

to get

2

12 210 \overline{∣ 25600} \underline{210} 460

Multiply the quotient digit

(2)

by the divisor

210

12 210 \overline{∣ 25600} \underline{210} 460 \underline{420}

Subtract

420

from

460

12 210 \overline{∣ 25600} \underline{210} 460 \underline{420} 40

Bring down the next number of the dividend

12 210 \overline{∣ 25600} \underline{210} 460 \underline{420} 400

12 210 \overline{∣ 25600} \underline{210} 460 \underline{420} 400

Divide

400

210

to get

1

Divide

400

210

to get

1

121 210 \overline{∣ 25600} \underline{210} 460 \underline{420} 400

Multiply the quotient digit

(1)

by the divisor

210

121 210 \overline{∣ 25600} \underline{210} 460 \underline{420} 400 \underline{210}

Subtract

210

from

400

121 210 \overline{∣ 25600} \underline{210} 460 \underline{420} 400 \underline{210} 190

121 210 \overline{∣ 25600} \underline{210} 460 \underline{420} 400 \underline{210} 190

The solution for Long Division of

\frac{25600}{210}

121

with remainder of

190

121 Remainder 190

= 190

= ((((19 0^{6} mod 210) \cdot 160) mod 210) \cdot 10) mod 210

Apply exponent rule:

a^{b c} = (a^{b})^{c}

= (((((19 0^{2})^{3} mod 210) \cdot 160) mod 210) \cdot 10) mod 210

Apply modulo rule:

(a^{b}) mod c = ((a mod c)^{b}) mod c

= ((((((19 0^{2} mod 210)^{3}) mod 210) \cdot 160) mod 210) \cdot 10) mod 210

19 0^{2} = 36100

= ((((((36100 mod 210)^{3}) mod 210) \cdot 160) mod 210) \cdot 10) mod 210

36100 mod 210 : 190

Find Remainder:

Long Division

\frac{36100}{210} : 171

Remainder

190

\frac{36100}{210}

Write the problem in long division format

210 \overline{∣ 36100}

Divide

361

210

to get

1

Divide

361

210

to get

1

1 210 \overline{∣ 36100}

Multiply the quotient digit

(1)

by the divisor

210

1 210 \overline{∣ 36100} \underline{210}

Subtract

210

from

361

1 210 \overline{∣ 36100} \underline{210} 151

Bring down the next number of the dividend

1 210 \overline{∣ 36100} \underline{210} 1510

1 210 \overline{∣ 36100} \underline{210} 1510

Divide

1510

210

to get

7

Divide

1510

210

to get

7

17 210 \overline{∣ 36100} \underline{210} 1510

Multiply the quotient digit

(7)

by the divisor

210

17 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470}

Subtract

1470

from

1510

17 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 40

Bring down the next number of the dividend

17 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 400

17 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 400

Divide

400

210

to get

1

Divide

400

210

to get

1

171 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 400

Multiply the quotient digit

(1)

by the divisor

210

171 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 400 \underline{210}

Subtract

210

from

400

171 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 400 \underline{210} 190

171 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 400 \underline{210} 190

The solution for Long Division of

\frac{36100}{210}

171

with remainder of

190

171 Remainder 190

= 190

= ((((19 0^{3} mod 210) \cdot 160) mod 210) \cdot 10) mod 210

Apply exponent rule:

a^{b + c} = a^{b} \cdot a^{c}

= ((((19 0^{2} \cdot 190 mod 210) \cdot 160) mod 210) \cdot 10) mod 210

Apply modulo rule:

(a \cdot b) mod c = ((a mod c) \cdot (b mod c)) mod c

= ((((((19 0^{2} mod 210) \cdot (190 mod 210)) mod 210) \cdot 160) mod 210) \cdot 10) mod 210

19 0^{2} = 36100

= ((((((36100 mod 210) \cdot (190 mod 210)) mod 210) \cdot 160) mod 210) \cdot 10) mod 210

190 mod 210 : 190

b > a > 0

then

a mod b = a

= 190

= ((((((36100 mod 210) \cdot 190) mod 210) \cdot 160) mod 210) \cdot 10) mod 210

36100 mod 210 : 190

Find Remainder:

Long Division

\frac{36100}{210} : 171

Remainder

190

\frac{36100}{210}

Write the problem in long division format

210 \overline{∣ 36100}

Divide

361

210

to get

1

Divide

361

210

to get

1

1 210 \overline{∣ 36100}

Multiply the quotient digit

(1)

by the divisor

210

1 210 \overline{∣ 36100} \underline{210}

Subtract

210

from

361

1 210 \overline{∣ 36100} \underline{210} 151

Bring down the next number of the dividend

1 210 \overline{∣ 36100} \underline{210} 1510

1 210 \overline{∣ 36100} \underline{210} 1510

Divide

1510

210

to get

7

Divide

1510

210

to get

7

17 210 \overline{∣ 36100} \underline{210} 1510

Multiply the quotient digit

(7)

by the divisor

210

17 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470}

Subtract

1470

from

1510

17 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 40

Bring down the next number of the dividend

17 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 400

17 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 400

Divide

400

210

to get

1

Divide

400

210

to get

1

171 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 400

Multiply the quotient digit

(1)

by the divisor

210

171 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 400 \underline{210}

Subtract

210

from

400

171 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 400 \underline{210} 190

171 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 400 \underline{210} 190

The solution for Long Division of

\frac{36100}{210}

171

with remainder of

190

171 Remainder 190

= 190

= ((((190 \cdot 190 mod 210) \cdot 160) mod 210) \cdot 10) mod 210

190 \cdot 190 = 36100

= ((((36100 mod 210) \cdot 160) mod 210) \cdot 10) mod 210

36100 mod 210 : 190

Find Remainder:

Long Division

\frac{36100}{210} : 171

Remainder

190

\frac{36100}{210}

Write the problem in long division format

210 \overline{∣ 36100}

Divide

361

210

to get

1

Divide

361

210

to get

1

1 210 \overline{∣ 36100}

Multiply the quotient digit

(1)

by the divisor

210

1 210 \overline{∣ 36100} \underline{210}

Subtract

210

from

361

1 210 \overline{∣ 36100} \underline{210} 151

Bring down the next number of the dividend

1 210 \overline{∣ 36100} \underline{210} 1510

1 210 \overline{∣ 36100} \underline{210} 1510

Divide

1510

210

to get

7

Divide

1510

210

to get

7

17 210 \overline{∣ 36100} \underline{210} 1510

Multiply the quotient digit

(7)

by the divisor

210

17 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470}

Subtract

1470

from

1510

17 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 40

Bring down the next number of the dividend

17 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 400

17 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 400

Divide

400

210

to get

1

Divide

400

210

to get

1

171 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 400

Multiply the quotient digit

(1)

by the divisor

210

171 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 400 \underline{210}

Subtract

210

from

400

171 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 400 \underline{210} 190

171 210 \overline{∣ 36100} \underline{210} 1510 \underline{1470} 400 \underline{210} 190

The solution for Long Division of

\frac{36100}{210}

171

with remainder of

190

171 Remainder 190

= 190

= ((190 \cdot 160 mod 210) \cdot 10) mod 210

190 \cdot 160 = 30400

= ((30400 mod 210) \cdot 10) mod 210

30400 mod 210 : 160

Find Remainder:

Long Division

\frac{30400}{210} : 144

Remainder

160

\frac{30400}{210}

Write the problem in long division format

210 \overline{∣ 30400}

Divide

304

210

to get

1

Divide

304

210

to get

1

1 210 \overline{∣ 30400}

Multiply the quotient digit

(1)

by the divisor

210

1 210 \overline{∣ 30400} \underline{210}

Subtract

210

from

304

1 210 \overline{∣ 30400} \underline{210} 94

Bring down the next number of the dividend

1 210 \overline{∣ 30400} \underline{210} 940

1 210 \overline{∣ 30400} \underline{210} 940

Divide

940

210

to get

4

Divide

940

210

to get

4

14 210 \overline{∣ 30400} \underline{210} 940

Multiply the quotient digit

(4)

by the divisor

210

14 210 \overline{∣ 30400} \underline{210} 940 \underline{840}

Subtract

840

from

940

14 210 \overline{∣ 30400} \underline{210} 940 \underline{840} 100

Bring down the next number of the dividend

14 210 \overline{∣ 30400} \underline{210} 940 \underline{840} 1000

14 210 \overline{∣ 30400} \underline{210} 940 \underline{840} 1000

Divide

1000

210

to get

4

Divide

1000

210

to get

4

144 210 \overline{∣ 30400} \underline{210} 940 \underline{840} 1000

Multiply the quotient digit

(4)

by the divisor

210

144 210 \overline{∣ 30400} \underline{210} 940 \underline{840} 1000 \underline{840}

Subtract

840

from

1000

144 210 \overline{∣ 30400} \underline{210} 940 \underline{840} 1000 \underline{840} 160

144 210 \overline{∣ 30400} \underline{210} 940 \underline{840} 1000 \underline{840} 160

The solution for Long Division of

\frac{30400}{210}

144

with remainder of

160

144 Remainder 160

= 160

= 160 \cdot 10 mod 210

160 \cdot 10 = 1600

= 1600 mod 210

1600 mod 210 : 130

Find Remainder:

Long Division

\frac{1600}{210} : 7

Remainder

130

\frac{1600}{210}

Write the problem in long division format

210 \overline{∣ 1600}

Divide

1600

210

to get

7

Divide

1600

210

to get

7

7 210 \overline{∣ 1600}

Multiply the quotient digit

(7)

by the divisor

210

7 210 \overline{∣ 1600} \underline{1470}

Subtract

1470

from

1600

7 210 \overline{∣ 1600} \underline{1470} 130

7 210 \overline{∣ 1600} \underline{1470} 130

The solution for Long Division of

\frac{1600}{210}

7

with remainder of

130

7 Remainder 130

= 130

= 130

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Frequently Asked Questions (FAQ)

What is 10^{40}\mod 210 ?
The solution to 10^{40}\mod 210 is 130