Number System - Study Mode

[#536] The numbers 1, 2, 3, 4, ......, 1000 are multiplied together. The number of zeros at the end (on the right) of the product must be :
Correct Answer

(D) 249

Explanation

Solution: Let N = 1 × 2 × 3 × 4 × ..... × 1000 = 1000! Clearly, the highest power of 2 in N very high as compared to that of 5. So, the number of zeros in N will be equal to the highest power of 5 in N. ∴ Required number of zeros = $$left[ {frac{{1000}}{5}}
ight] + left[ {frac{{1000}}{{{5^2}}}}
ight] + left[ {frac{{1000}}{{{5^3}}}}
ight]$$ xa0 xa0xa0 $$ + left[ {frac{{1000}}{{{5^4}}}}
ight]$$ = 200 + 40 + 8 + 1 = 249

[#537] The number 89715938* is divisible by 4. The unknown non-zero digit marked as * will be :
Correct Answer

(C) 4

Explanation

Solution: Let the missing digit be x. Then, (80 + x) must be divisible by 4. Hence, x = 4

[#538] If p is a prime number greater than 3, then (p 2 - 1) is always divisible by :
Correct Answer

(C) 24

Explanation

Solution: ⇔ p = 5 ⇒ (p 2 - 1) = (25 - 1) ⇒ (p 2 - 1) = 24, which is divisible by 24 ⇔ p = 7 ⇒ (p 2 - 1) = (49 - 1) ⇒ (p 2 - 1) = 48, which is divisible by 24 ⇔ p = 11 ⇒ (p 2 - 1) = (121 - 1) ⇒ (p 2 - 1) = 120, which is divisible by 24 Hence, (p 2 - 1) is always divisible by 24

[#539] The unit's digit of 13 2003 is :
Correct Answer

(C) 7

Explanation

Solution: 3 4 gives unit digit 1 So, (3 4 ) 500 gives unit digit 1 And, 3 3 gives unit digit 7 ∴ (13) 2003 gives unit digit = (1 × 7) = 7

[#540] What is the unit digit in {(6374) 1793 x (625) 317 x (341 491 )}?
Correct Answer

(A) 0

Explanation

Solution: Unit digit in (6374) 1793 = Unit digit in (4) 1793 = Unit digit in [(4 2 ) 896 x 4] = Unit digit in (6 x 4) = 4 Unit digit in (625) 317 = Unit digit in (5) 317 = 5 Unit digit in (341) 491 = Unit digit in (1) 491 = 1 Required digit = Unit digit in (4 x 5 x 1) = 0