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QUESTION
Q
When numbers are written in base b, we have 15 × 22 = 414, the value of b is ?
When numbers are written in base b, we have 15 × 22 = 414, the value of b is ?
A
None of these
B
4
C
8
D
7
E
6
A
None of these
B
4
C
8
D
7
E
6
Solution
The numbers when written in base b.15 = 1(b1) + 5(b0) = b + 522 = 2(b1) + 2(b0) = 2b + 2414 = 4(b2) + 1(b1) + 4(b0) = 4b2 + 1b + 415...
The numbers when written in base b.
15 = 1(b1) + 5(b0) = b + 5
22 = 2(b1) + 2(b0) = 2b + 2
414 = 4(b2) + 1(b1) + 4(b0) = 4b2 + 1b + 4
15 × 22 = 414
(b + 5) × (2b + 2) = 4b2 + 1b + 4
2b2 + 12b + 10 = 4b2 + 1b + 4
2b2 − 11b − 6 = 0
(b − 6) (2b +1) = 0
b = 6 or −1/2
Therefore b = 6 (since b cannot be negative)
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Keynote
ABCDE written to the base n = A(n4) + B(n3) + C(n2) + D(n1) + E(n0)12345 to the base 2 = 1(24) + 2(23) + 3(22) + 4(21)...
ABCDE written to the base n = A(n4) + B(n3) + C(n2) + D(n1) + E(n0)
12345 to the base 2 = 1(24) + 2(23) + 3(22) + 4(21) + 5(20)
= 1(16) + 2(8) + 3(4) + 4(2) + 5(1)
= 57
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