Find the lowest number which leaves 4 as remainder when divide by 9 and 12 . Find the lowest number which being increased by 3 is exactly divided by 8,12 and 16 . Find the lowest number which is less by 5 to be divided by 15,25 and 50 exactly. Find the lowest number which is less by 2 to be divided by 56 and 98 exactly. The product of the L.C.M. and H.C.F. of two numbers is 80 . If one of the numbers is ind the other number. ind the lowest number which is less by 9 to be divided by 21,35 and 49 exactly. he product of two numbers is 192 . If the H.C.F. of the numbers is 4 , find their L.C.)

Sure, let's tackle each of your problems step by step. --- ### 1. **Find the lowest number which leaves 4 as remainder when divided by 9 and 12.** **Solution:** We need to find the smallest number \( N \) such that: - \( N \mod 9 = 4 \) - \( N \mod 12 = 4 \) This implies that \( N - 4 \) is divisible by both 9 and 12. 1. **Find the Least Common Multiple (LCM) of 9 and 12:** - Prime factors of 9: \( 3^2 \) - Prime factors of 12: \( 2^2 \times 3 \) LCM = \( 2^2 \times 3^2 = 4 \times 9 = 36 \) 2. **Calculate \( N \):** \[ N = \text{LCM}(9, 12) + 4 = 36 + 4 = 40 \] **Answer:** The lowest number is **40**. --- ### 2. **Find the lowest number which, when increased by 3, is exactly divisible by 8, 12, and 16.** **Solution:** Let \( N + 3 \) be divisible by 8, 12, and 16. 1. **Find the LCM of 8, 12, and 16:** - Prime factors of 8: \( 2^3 \) - Prime factors of 12: \( 2^2 \times 3 \) - Prime factors of 16: \( 2^4 \) LCM = \( 2^4 \times 3 = 16 \times 3 = 48 \) 2. **Calculate \( N \):** \[ N + 3 = 48 \implies N = 48 - 3 = 45 \] **Answer:** The lowest number is **45**. --- ### 3. **Find the lowest number which is less by 5 to be divided by 15, 25, and 50 exactly.** **Solution:** We need to find the smallest number \( N \) such that \( N + 5 \) is divisible by 15, 25, and 50. 1. **Find the LCM of 15, 25, and 50:** - Prime factors of 15: \( 3 \times 5 \) - Prime factors of 25: \( 5^2 \) - Prime factors of 50: \( 2 \times 5^2 \) LCM = \( 2 \times 3 \times 5^2 = 2 \times 3 \times 25 = 150 \) 2. **Calculate \( N \):** \[ N + 5 = 150 \implies N = 150 - 5 = 145 \] **Answer:** The lowest number is **145**. --- ### 4. **Find the lowest number which is less by 2 to be divided by 56 and 98 exactly.** **Solution:** We need to find the smallest number \( N \) such that \( N + 2 \) is divisible by 56 and 98. 1. **Find the LCM of 56 and 98:** - Prime factors of 56: \( 2^3 \times 7 \) - Prime factors of 98: \( 2 \times 7^2 \) LCM = \( 2^3 \times 7^2 = 8 \times 49 = 392 \) 2. **Calculate \( N \):** \[ N + 2 = 392 \implies N = 392 - 2 = 390 \] **Answer:** The lowest number is **390**. --- ### 5. **The product of the L.C.M. and H.C.F. of two numbers is 80. If one of the numbers is ___, find the other number.** **Solution:** It seems like there's some information missing in your question, specifically the value of one of the numbers. To solve such problems, we typically use the relationship: \[ \text{Product of two numbers} = \text{HCF} \times \text{LCM} \] Given: \[ \text{HCF} \times \text{LCM} = 80 \] Without knowing at least one of the numbers or additional information, we cannot determine the other number. Could you please provide the value of one of the numbers or any additional details? --- ### 6. **Find the lowest number which is less by 9 to be divided by 21, 35, and 49 exactly.** **Solution:** We need to find the smallest number \( N \) such that \( N + 9 \) is divisible by 21, 35, and 49. 1. **Find the LCM of 21, 35, and 49:** - Prime factors of 21: \( 3 \times 7 \) - Prime factors of 35: \( 5 \times 7 \) - Prime factors of 49: \( 7^2 \) LCM = \( 3 \times 5 \times 7^2 = 3 \times 5 \times 49 = 735 \) 2. **Calculate \( N \):** \[ N + 9 = 735 \implies N = 735 - 9 = 726 \] **Answer:** The lowest number is **726**. --- ### 7. **The product of two numbers is 192. If the H.C.F. of the numbers is 4, find their L.C.M.** **Solution:** Let the two numbers be \( a \) and \( b \). We know that: \[ a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b) \] Given: \[ a \times b = 192 \\ \text{HCF}(a, b) = 4 \] Plugging into the formula: \[ 4 \times \text{LCM}(a, b) = 192 \\ \text{LCM}(a, b) = \frac{192}{4} = 48 \] **Answer:** The L.C.M. of the numbers is **48**. --- If you have any further questions or need clarification on any of the solutions, feel free to ask!